diff --git a/mlir/include/mlir/IR/AffineExpr.h b/mlir/include/mlir/IR/AffineExpr.h
index 4d8516c1aa01..b5bf8f2f53f0 100644
--- a/mlir/include/mlir/IR/AffineExpr.h
+++ b/mlir/include/mlir/IR/AffineExpr.h
@@ -23,12 +23,229 @@
 #ifndef MLIR_IR_AFFINE_EXPR_H
 #define MLIR_IR_AFFINE_EXPR_H
 
+#include "mlir/Support/LLVM.h"
+
 namespace mlir {
 
-class AffineExpr  {
+class MLIRContext;
+
+/// A one-dimensional affine expression.
+/// AffineExpression's are immutable (like Type's)
+class AffineExpr {
  public:
-  AffineExpr();
-  // TODO(andydavis,bondhugula) Implement affine expressions.
+  enum class Kind {
+    // Add.
+    Add,
+    // Mul.
+    Mul,
+    // Mod.
+    Mod,
+    // Floordiv
+    FloorDiv,
+    // Ceildiv
+    CeilDiv,
+
+    /// This is a marker for the last affine binary op. The range of binary op's
+    /// is expected to be this element and earlier.
+    LAST_AFFINE_BINARY_OP = CeilDiv,
+
+    // Unary op negation
+    Neg,
+
+    // Constant integer.
+    Constant,
+    // Dimensional identifier.
+    DimId,
+    // Symbolic identifier.
+    SymbolId,
+  };
+
+  /// Return the classification for this type.
+  Kind getKind() const { return kind; }
+
+  ~AffineExpr() = default;
+
+  void print(raw_ostream &os) const;
+  void dump() const;
+
+ protected:
+  explicit AffineExpr(Kind kind) : kind(kind) {}
+
+ private:
+  /// Classification of the subclass
+  const Kind kind;
+};
+
+/// Binary affine expression.
+class AffineBinaryOpExpr : public AffineExpr {
+ public:
+  static AffineBinaryOpExpr *get(Kind kind, AffineExpr *lhsOperand,
+                                 AffineExpr *rhsOperand, MLIRContext *context);
+
+  AffineExpr *getLeftOperand() const { return lhsOperand; }
+  AffineExpr *getRightOperand() const { return rhsOperand; }
+
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() <= Kind::LAST_AFFINE_BINARY_OP;
+  }
+
+ protected:
+  explicit AffineBinaryOpExpr(Kind kind, AffineExpr *lhsOperand,
+                              AffineExpr *rhsOperand)
+      : AffineExpr(kind), lhsOperand(lhsOperand), rhsOperand(rhsOperand) {}
+
+  AffineExpr *const lhsOperand;
+  AffineExpr *const rhsOperand;
+};
+
+/// Binary affine add expression.
+class AffineAddExpr : public AffineBinaryOpExpr {
+ public:
+  static AffineAddExpr *get(AffineExpr *lhsOperand, AffineExpr *rhsOperand,
+                            MLIRContext *context);
+
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::Add;
+  }
+
+ private:
+  explicit AffineAddExpr(AffineExpr *lhsOperand, AffineExpr *rhsOperand)
+      : AffineBinaryOpExpr(Kind::Add, lhsOperand, rhsOperand) {}
+};
+
+/// Binary affine mul expression.
+class AffineMulExpr : public AffineBinaryOpExpr {
+ public:
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::Mul;
+  }
+
+ private:
+  explicit AffineMulExpr(AffineExpr *lhsOperand, AffineExpr *rhsOperand)
+      : AffineBinaryOpExpr(Kind::Mul, lhsOperand, rhsOperand) {}
+};
+
+/// Binary affine mod expression.
+class AffineModExpr : public AffineBinaryOpExpr {
+ public:
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::Mod;
+  }
+
+ private:
+  explicit AffineModExpr(AffineExpr *lhsOperand, AffineExpr *rhsOperand)
+      : AffineBinaryOpExpr(Kind::Mod, lhsOperand, rhsOperand) {}
+};
+
+/// Binary affine floordiv expression.
+class AffineFloorDivExpr : public AffineBinaryOpExpr {
+ public:
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::FloorDiv;
+  }
+
+ private:
+  explicit AffineFloorDivExpr(AffineExpr *lhsOperand, AffineExpr *rhsOperand)
+      : AffineBinaryOpExpr(Kind::FloorDiv, lhsOperand, rhsOperand) {}
+};
+
+/// Binary affine ceildiv expression.
+class AffineCeilDivExpr : public AffineBinaryOpExpr {
+ public:
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::CeilDiv;
+  }
+
+ private:
+  explicit AffineCeilDivExpr(AffineExpr *lhsOperand, AffineExpr *rhsOperand)
+      : AffineBinaryOpExpr(Kind::CeilDiv, lhsOperand, rhsOperand) {}
+};
+
+/// Unary affine expression.
+class AffineUnaryOpExpr : public AffineExpr {
+ public:
+  static AffineUnaryOpExpr *get(const AffineExpr &operand,
+                                MLIRContext *context);
+
+  static AffineUnaryOpExpr *get(const AffineExpr &operand);
+  AffineExpr *getOperand() const { return operand; }
+
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::Neg;
+  }
+
+ private:
+  explicit AffineUnaryOpExpr(Kind kind, AffineExpr *operand)
+      : AffineExpr(kind), operand(operand) {}
+
+  AffineExpr *operand;
+};
+
+/// A argument identifier appearing in an affine expression
+class AffineDimExpr : public AffineExpr {
+ public:
+  static AffineDimExpr *get(unsigned position, MLIRContext *context);
+
+  unsigned getPosition() const { return position; }
+
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::DimId;
+  }
+
+ private:
+  explicit AffineDimExpr(unsigned position)
+      : AffineExpr(Kind::DimId), position(position) {}
+
+  /// Position of this identifier in the argument list.
+  unsigned position;
+};
+
+/// A symbolic identifier appearing in an affine expression
+class AffineSymbolExpr : public AffineExpr {
+ public:
+  static AffineSymbolExpr *get(unsigned position, MLIRContext *context);
+
+  unsigned getPosition() const { return position; }
+
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::SymbolId;
+  }
+
+ private:
+  explicit AffineSymbolExpr(unsigned position)
+      : AffineExpr(Kind::SymbolId), position(position) {}
+
+  /// Position of this identifier in the symbol list.
+  unsigned position;
+};
+
+/// An integer constant appearing in affine expression.
+class AffineConstantExpr : public AffineExpr {
+ public:
+  static AffineConstantExpr *get(int64_t constant, MLIRContext *context);
+
+  int64_t getValue() const { return constant; }
+
+  /// Methods for support type inquiry through isa, cast, and dyn_cast.
+  static bool classof(const AffineExpr *expr) {
+    return expr->getKind() == Kind::Constant;
+  }
+
+ private:
+  explicit AffineConstantExpr(int64_t constant)
+      : AffineExpr(Kind::Constant), constant(constant) {}
+
+  // The constant.
+  int64_t constant;
 };
 
 } // end namespace mlir
diff --git a/mlir/include/mlir/IR/AffineMap.h b/mlir/include/mlir/IR/AffineMap.h
index af1df9687fe0..626605059a89 100644
--- a/mlir/include/mlir/IR/AffineMap.h
+++ b/mlir/include/mlir/IR/AffineMap.h
@@ -26,30 +26,43 @@
 #include <vector>
 
 #include "mlir/Support/LLVM.h"
+#include "llvm/ADT/ArrayRef.h"
 
 namespace mlir {
 
+class MLIRContext;
 class AffineExpr;
 
-class AffineMap  {
+/// A multi-dimensional affine map
+/// Affine map's are immutable like Type's, and they are uniqued.
+/// Eg: (d0, d1) -> (d0/128, d0 mod 128, d1)
+/// The names used (d0, d1) don't matter - it's the mathematical function that
+/// is unique to this affine map.
+class AffineMap {
  public:
-  // Constructs an AffineMap with 'dimCount' dimension identifiers, and
-  // 'symbolCount' symbols.
-  // TODO(andydavis) Pass in ArrayRef<AffineExpr*> to populate list of exprs.
-  AffineMap(unsigned dimCount, unsigned symbolCount);
+  static AffineMap *get(unsigned dimCount, unsigned symbolCount,
+                        ArrayRef<AffineExpr *> exprs,
+                        MLIRContext *context);
 
   // Prints affine map to 'os'.
   void print(raw_ostream &os) const;
+  void dump() const;
+
+  unsigned dimCount() const { return numDims; }
+  unsigned symbolCount() const { return numSymbols; }
 
  private:
-  // Number of dimensional indentifiers.
-  const unsigned dimCount;
-  // Number of symbols.
-  const unsigned symbolCount;
-  // TODO(andydavis) Do not use std::vector here (array size is not dynamic).
-  std::vector<AffineExpr*> exprs;
+  AffineMap(unsigned dimCount, unsigned symbolCount,
+            ArrayRef<AffineExpr *> exprs);
+
+  const unsigned numDims;
+  const unsigned numSymbols;
+
+  /// The affine expressions for this (multi-dimensional) map.
+  /// TODO: use trailing objects for these
+  ArrayRef<AffineExpr *> exprs;
 };
 
-} // end namespace mlir
+}  // end namespace mlir
 
 #endif  // MLIR_IR_AFFINE_MAP_H
diff --git a/mlir/include/mlir/IR/Module.h b/mlir/include/mlir/IR/Module.h
index bf7707812a94..7d22efd96e20 100644
--- a/mlir/include/mlir/IR/Module.h
+++ b/mlir/include/mlir/IR/Module.h
@@ -22,10 +22,13 @@
 #ifndef MLIR_IR_MODULE_H
 #define MLIR_IR_MODULE_H
 
+#include "mlir/IR/AffineMap.h"
 #include "mlir/IR/Function.h"
-#include <vector>
 
 namespace mlir {
+
+class AffineMap;
+
 class Module {
 public:
   explicit Module();
@@ -33,6 +36,9 @@ public:
   // FIXME: wrong representation and API.
   std::vector<Function*> functionList;
 
+  // FIXME: wrong representation and API.
+  // These affine maps are immutable
+  std::vector<const AffineMap *> affineMapList;
 
   void print(raw_ostream &os) const;
   void dump() const;
diff --git a/mlir/lib/IR/AffineExpr.cpp b/mlir/lib/IR/AffineExpr.cpp
index 63d8d7df2c38..87901f03c77a 100644
--- a/mlir/lib/IR/AffineExpr.cpp
+++ b/mlir/lib/IR/AffineExpr.cpp
@@ -18,6 +18,3 @@
 #include "mlir/IR/AffineExpr.h"
 
 using namespace mlir;
-
-AffineExpr::AffineExpr() {
-}
diff --git a/mlir/lib/IR/AffineMap.cpp b/mlir/lib/IR/AffineMap.cpp
index 83d1d2359cf7..863175124708 100644
--- a/mlir/lib/IR/AffineMap.cpp
+++ b/mlir/lib/IR/AffineMap.cpp
@@ -17,15 +17,13 @@
 
 #include "mlir/IR/AffineMap.h"
 #include "llvm/ADT/StringRef.h"
-#include "llvm/Support/raw_ostream.h"
 
 using namespace mlir;
 
-AffineMap::AffineMap(unsigned dimCount, unsigned symbolCount)
-    : dimCount(dimCount), symbolCount(symbolCount) {
-}
-
-void AffineMap::print(raw_ostream &os) const {
-  // TODO(andydavis) Print out affine map based on dimensionCount and
-  // symbolCount: (d0, d1) [S0, S1] -> (d0 + S0, d1 + S1)
+// TODO(clattner):  make this ctor take an LLVMContext.  This will eventually
+// copy the elements into the context.
+AffineMap::AffineMap(unsigned dimCount, unsigned symbolCount,
+                     ArrayRef<AffineExpr *> exprs)
+  : numDims(dimCount), numSymbols(symbolCount), exprs(exprs) {
+    // TODO(bondhugula)
 }
diff --git a/mlir/lib/IR/AsmPrinter.cpp b/mlir/lib/IR/AsmPrinter.cpp
index 639000b53d30..067697cdf064 100644
--- a/mlir/lib/IR/AsmPrinter.cpp
+++ b/mlir/lib/IR/AsmPrinter.cpp
@@ -20,13 +20,15 @@
 //
 //===----------------------------------------------------------------------===//
 
+#include "mlir/IR/AffineExpr.h"
+#include "mlir/IR/AffineMap.h"
 #include "mlir/IR/CFGFunction.h"
 #include "mlir/IR/MLFunction.h"
 #include "mlir/IR/Module.h"
 #include "mlir/IR/Types.h"
 #include "mlir/Support/STLExtras.h"
-#include "llvm/Support/raw_ostream.h"
 #include "llvm/ADT/DenseMap.h"
+#include "llvm/Support/raw_ostream.h"
 using namespace mlir;
 
 
@@ -162,6 +164,15 @@ void Instruction::dump() const {
   print(llvm::errs());
 }
 
+void AffineExpr::print(raw_ostream &os) const {
+  // TODO(bondhugula): print out affine expression
+}
+
+void AffineMap::print(raw_ostream &os) const {
+  // TODO(andydavis) Print out affine map based on dimensionCount and
+  // symbolCount: (d0, d1) [S0, S1] -> (d0 + S0, d1 + S1)
+}
+
 void BasicBlock::print(raw_ostream &os) const {
   CFGFunctionState state(getFunction(), os);
   state.print();
@@ -208,6 +219,8 @@ void MLFunction::print(raw_ostream &os) const {
 }
 
 void Module::print(raw_ostream &os) const {
+  for (auto *map : affineMapList)
+    map->print(os);
   for (auto *fn : functionList)
     fn->print(os);
 }
diff --git a/mlir/lib/IR/MLIRContext.cpp b/mlir/lib/IR/MLIRContext.cpp
index 5f2bd8ec7640..7c1112bc2300 100644
--- a/mlir/lib/IR/MLIRContext.cpp
+++ b/mlir/lib/IR/MLIRContext.cpp
@@ -17,6 +17,8 @@
 
 #include "mlir/IR/MLIRContext.h"
 #include "mlir/IR/Identifier.h"
+#include "mlir/IR/AffineExpr.h"
+#include "mlir/IR/AffineMap.h"
 #include "mlir/IR/Types.h"
 #include "mlir/Support/LLVM.h"
 #include "llvm/ADT/DenseSet.h"
@@ -44,6 +46,23 @@ struct FunctionTypeKeyInfo : DenseMapInfo<FunctionType*> {
     return lhs == KeyTy(rhs->getInputs(), rhs->getResults());
   }
 };
+struct AffineMapKeyInfo : DenseMapInfo<AffineMap *> {
+  // Affine maps are uniqued based on their arguments and affine expressions
+  using KeyTy = std::pair<unsigned, unsigned>;
+  using DenseMapInfo<AffineMap *>::getHashValue;
+  using DenseMapInfo<AffineMap *>::isEqual;
+
+  static unsigned getHashValue(KeyTy key) {
+    // FIXME(bondhugula): placeholder for now
+    return hash_combine(key.first, key.second);
+  }
+
+  static bool isEqual(const KeyTy &lhs, const FunctionType *rhs) {
+    // TODO(bondhugula)
+    return false;
+  }
+};
+
 struct VectorTypeKeyInfo : DenseMapInfo<VectorType*> {
   // Vectors are uniqued based on their element type and shape.
   using KeyTy = std::pair<Type*, ArrayRef<unsigned>>;
@@ -97,6 +116,10 @@ public:
   // Primitive type uniquing.
   PrimitiveType *primitives[int(TypeKind::LAST_PRIMITIVE_TYPE)+1] = { nullptr };
 
+  // Affine map uniquing.
+  using AffineMapSet = DenseSet<AffineMap *, AffineMapKeyInfo>;
+  AffineMapSet affineMaps;
+
   /// Function type uniquing.
   using FunctionTypeSet = DenseSet<FunctionType*, FunctionTypeKeyInfo>;
   FunctionTypeSet functions;
@@ -316,3 +339,52 @@ UnrankedTensorType *UnrankedTensorType::get(Type *elementType) {
   // Cache and return it.
   return existing.first->second = result;
 }
+
+// TODO(bondhugula,andydavis): unique affine maps based on dim list,
+// symbol list and all affine expressions contained
+AffineMap *AffineMap::get(unsigned dimCount,
+                          unsigned symbolCount,
+                          ArrayRef<AffineExpr *> exprs,
+                          MLIRContext *context) {
+  // TODO(bondhugula)
+  return new AffineMap(dimCount, symbolCount, exprs);
+}
+
+AffineBinaryOpExpr *AffineBinaryOpExpr::get(AffineExpr::Kind kind,
+                                            AffineExpr *lhsOperand,
+                                            AffineExpr *rhsOperand,
+                                            MLIRContext *context) {
+  // TODO(bondhugula): allocate this through context
+  // FIXME
+  return new AffineBinaryOpExpr(kind, lhsOperand, rhsOperand);
+}
+
+AffineAddExpr *AffineAddExpr::get(AffineExpr *lhsOperand,
+                                  AffineExpr *rhsOperand,
+                                  MLIRContext *context) {
+  // TODO(bondhugula): allocate this through context
+  // FIXME
+  return new AffineAddExpr(lhsOperand, rhsOperand);
+}
+
+// TODO(bondhugula): add functions for AffineMulExpr, mod, floordiv, ceildiv
+
+AffineDimExpr *AffineDimExpr::get(unsigned position, MLIRContext *context) {
+  // TODO(bondhugula): complete this
+  // FIXME: this should be POD
+  return new AffineDimExpr(position);
+}
+
+AffineSymbolExpr *AffineSymbolExpr::get(unsigned position,
+                                        MLIRContext *context) {
+  // TODO(bondhugula): complete this
+  // FIXME: this should be POD
+  return new AffineSymbolExpr(position);
+}
+
+AffineConstantExpr *AffineConstantExpr::get(int64_t constant,
+                                            MLIRContext *context) {
+  // TODO(bondhugula): complete this
+  // FIXME: this should be POD
+  return new AffineConstantExpr(constant);
+}
diff --git a/mlir/lib/Parser/Lexer.cpp b/mlir/lib/Parser/Lexer.cpp
index e3e5e9ec358d..17755e0291f9 100644
--- a/mlir/lib/Parser/Lexer.cpp
+++ b/mlir/lib/Parser/Lexer.cpp
@@ -78,9 +78,14 @@ Token Lexer::lexToken() {
   case ')': return formToken(Token::r_paren, tokStart);
   case '{': return formToken(Token::l_brace, tokStart);
   case '}': return formToken(Token::r_brace, tokStart);
+  case '[': return formToken(Token::l_bracket, tokStart);
+  case ']': return formToken(Token::r_bracket, tokStart);
   case '<': return formToken(Token::less, tokStart);
   case '>': return formToken(Token::greater, tokStart);
+  case '=': return formToken(Token::equal, tokStart);
 
+  case '+': return formToken(Token::plus, tokStart);
+  case '*': return formToken(Token::star, tokStart);
   case '-':
     if (*curPtr == '>') {
       ++curPtr;
@@ -246,3 +251,4 @@ Token Lexer::lexString(const char *tokStart) {
     }
   }
 }
+
diff --git a/mlir/lib/Parser/Parser.cpp b/mlir/lib/Parser/Parser.cpp
index c62ee5d14f63..692705018aa8 100644
--- a/mlir/lib/Parser/Parser.cpp
+++ b/mlir/lib/Parser/Parser.cpp
@@ -18,12 +18,14 @@
 // This file implements the parser for the MLIR textual form.
 //
 //===----------------------------------------------------------------------===//
+#include <stack>
 
 #include "mlir/Parser.h"
 #include "Lexer.h"
+#include "mlir/IR/AffineExpr.h"
 #include "mlir/IR/AffineMap.h"
-#include "mlir/IR/Module.h"
 #include "mlir/IR/CFGFunction.h"
+#include "mlir/IR/Module.h"
 #include "mlir/IR/MLFunction.h"
 #include "mlir/IR/Types.h"
 #include "llvm/Support/SourceMgr.h"
@@ -33,6 +35,7 @@ using llvm::SMLoc;
 
 namespace {
 class CFGFunctionParserState;
+class AffineMapParserState;
 
 /// Simple enum to make code read better in cases that would otherwise return a
 /// bool value.  Failure is "true" in a boolean context.
@@ -128,8 +131,19 @@ private:
   Type *parseType();
   ParseResult parseTypeList(SmallVectorImpl<Type*> &elements);
 
+  // Identifiers
+  ParseResult parseDimIdList(SmallVectorImpl<StringRef> &dims,
+                             SmallVectorImpl<StringRef> &symbols);
+  ParseResult parseSymbolIdList(SmallVectorImpl<StringRef> &dims,
+                                SmallVectorImpl<StringRef> &symbols);
+  StringRef parseDimOrSymbolId(SmallVectorImpl<StringRef> &dims,
+                               SmallVectorImpl<StringRef> &symbols,
+                               bool symbol);
+
   // Polyhedral structures
   ParseResult parseAffineMapDef();
+  AffineMap *parseAffineMapInline(StringRef mapId);
+  AffineExpr *parseAffineExpr(AffineMapParserState &state);
 
   // Functions.
   ParseResult parseFunctionSignature(StringRef &name, FunctionType *&type);
@@ -476,6 +490,40 @@ ParseResult Parser::parseTypeList(SmallVectorImpl<Type*> &elements) {
   return ParseSuccess;
 }
 
+namespace {
+/// This class represents the transient parser state while parsing an affine
+/// expression.
+class AffineMapParserState {
+ public:
+  explicit AffineMapParserState(ArrayRef<StringRef> dims,
+                                ArrayRef<StringRef> symbols) :
+    dims_(dims), symbols_(symbols) {}
+
+  unsigned dimCount() const { return dims_.size(); }
+  unsigned symbolCount() const { return symbols_.size(); }
+
+  // Stack operations for affine expression parsing
+  // TODO(bondhugula): all of this will be improved/made more principled
+  void pushAffineExpr(AffineExpr *expr) { exprStack.push(expr); }
+  AffineExpr *popAffineExpr() {
+    auto *t = exprStack.top();
+    exprStack.pop();
+    return t;
+  }
+  AffineExpr *topAffineExpr() { return exprStack.top(); }
+
+  ArrayRef<StringRef> getDims() const { return dims_; }
+  ArrayRef<StringRef> getSymbols() const { return symbols_; }
+
+ private:
+  const ArrayRef<StringRef> dims_;
+  const ArrayRef<StringRef> symbols_;
+
+  // TEMP: stack to hold affine expressions
+  std::stack<AffineExpr *> exprStack;
+};
+}  // end anonymous namespace
+
 //===----------------------------------------------------------------------===//
 // Polyhedral structures.
 //===----------------------------------------------------------------------===//
@@ -483,24 +531,205 @@ ParseResult Parser::parseTypeList(SmallVectorImpl<Type*> &elements) {
 /// Affine map declaration.
 ///
 ///  affine-map-def ::= affine-map-id `=` affine-map-inline
-///  affine-map-inline ::= dim-and-symbol-id-lists `->` multi-dim-affine-expr
-///                        ( `size` `(` dim-size (`,` dim-size)* `)` )?
-///  dim-size ::= affine-expr | `min` `(` affine-expr ( `,` affine-expr)+ `)`
 ///
 ParseResult Parser::parseAffineMapDef() {
   assert(curToken.is(Token::affine_map_identifier));
 
   StringRef affineMapId = curToken.getSpelling().drop_front();
+  consumeToken(Token::affine_map_identifier);
+
   // Check that 'affineMapId' is unique.
   // TODO(andydavis) Add a unit test for this case.
   if (affineMaps.count(affineMapId) > 0)
     return emitError("redefinition of affine map id '" + affineMapId + "'");
+  // Parse the '='
+  if (!consumeIf(Token::equal))
+    return emitError("expected '=' in affine map outlined definition");
 
-  consumeToken(Token::affine_map_identifier);
+  auto *affineMap = parseAffineMapInline(affineMapId);
+  affineMaps[affineMapId].reset(affineMap);
+  if (!affineMap) return ParseFailure;
 
-  // TODO(andydavis,bondhugula) Parse affine map definition.
-  affineMaps[affineMapId].reset(new AffineMap(1, 0));
-  return ParseSuccess;
+  module->affineMapList.push_back(affineMap);
+  return affineMap ? ParseSuccess : ParseFailure;
+}
+
+///
+/// Parse a multi-dimensional affine expression
+/// affine-expr ::= `(` affine-expr `)`
+///              | affine-expr `+` affine-expr
+///              | affine-expr `-` affine-expr
+///              | `-`? integer-literal `*` affine-expr
+///              | `ceildiv` `(` affine-expr `,` integer-literal `)`
+///              | `floordiv` `(` affine-expr `,` integer-literal `)`
+///              | affine-expr `mod` integer-literal
+///              | bare-id
+///              | `-`? integer-literal
+/// multi-dim-affine-expr ::= `(` affine-expr (`,` affine-expr)* `)
+///
+/// Use 'state' to check if valid identifiers appear.
+///
+AffineExpr *Parser::parseAffineExpr(AffineMapParserState &state) {
+  // TODO(bondhugula): complete support for this
+  // The code below is all placeholder / it is wrong / not complete
+  // Operator precedence not considered; pure left to right associativity
+  if (curToken.is(Token::comma)) {
+    emitError("expecting affine expression");
+    return nullptr;
+  }
+
+  while (curToken.isNot(Token::comma, Token::r_paren,
+                        Token::eof, Token::error)) {
+    switch (curToken.getKind()) {
+      case Token::bare_identifier: {
+        // TODO(bondhugula): look up state to see if it's a symbol or dim_id and
+        // get its position
+        AffineExpr *expr = AffineDimExpr::get(0, context);
+        state.pushAffineExpr(expr);
+        consumeToken(Token::bare_identifier);
+        break;
+      }
+      case Token::plus: {
+        consumeToken(Token::plus);
+        if (state.topAffineExpr()) {
+          auto lChild = state.popAffineExpr();
+          auto rChild = parseAffineExpr(state);
+          if (rChild) {
+            auto binaryOpExpr = AffineAddExpr::get(lChild, rChild, context);
+            state.popAffineExpr();
+            state.pushAffineExpr(binaryOpExpr);
+          } else {
+            emitError("right operand of + missing");
+          }
+        } else {
+          emitError("left operand of + missing");
+        }
+        break;
+      }
+      case Token::integer: {
+        AffineExpr *expr = AffineConstantExpr::get(
+            curToken.getUnsignedIntegerValue().getValue(), context);
+        state.pushAffineExpr(expr);
+        consumeToken(Token::integer);
+        break;
+      }
+      case Token::l_paren: {
+        consumeToken(Token::l_paren);
+        break;
+      }
+      case Token::r_paren: {
+        consumeToken(Token::r_paren);
+        break;
+      }
+      default: {
+        emitError("affine map expr parse impl incomplete/unexpected token");
+        return nullptr;
+      }
+    }
+  }
+  if (!state.topAffineExpr()) {
+    // An error will be emitted by parse comma separated list on an empty list
+    return nullptr;
+  }
+  return state.topAffineExpr();
+}
+
+// Return empty string if no bare id was found
+StringRef Parser::parseDimOrSymbolId(SmallVectorImpl<StringRef> &dims,
+                                     SmallVectorImpl<StringRef> &symbols,
+                                     bool symbol = false) {
+  if (curToken.isNot(Token::bare_identifier)) {
+    emitError("expected bare identifier");
+    return StringRef();
+  }
+  // TODO(bondhugula): check whether the id already exists in either
+  // state.symbols or state.dims; report error if it does; otherwise create a
+  // new one.
+  StringRef ref = curToken.getSpelling();
+  consumeToken(Token::bare_identifier);
+  return ref;
+}
+
+ParseResult Parser::parseSymbolIdList(SmallVectorImpl<StringRef> &dims,
+                                      SmallVectorImpl<StringRef> &symbols) {
+  if (!consumeIf(Token::l_bracket)) return emitError("expected '['");
+
+  auto parseElt = [&]() -> ParseResult {
+    auto elt = parseDimOrSymbolId(dims, symbols, true);
+    // FIXME(bondhugula): assuming dim arg for now
+    if (!elt.empty()) {
+      symbols.push_back(elt);
+      return ParseSuccess;
+    }
+    return ParseFailure;
+  };
+  return parseCommaSeparatedList(Token::r_bracket, parseElt);
+}
+
+// TODO(andy,bondhugula)
+ParseResult Parser::parseDimIdList(SmallVectorImpl<StringRef> &dims,
+                                   SmallVectorImpl<StringRef> &symbols) {
+  if (!consumeIf(Token::l_paren))
+    return emitError("expected '(' at start of dimensional identifiers list");
+
+  auto parseElt = [&]() -> ParseResult {
+    auto elt = parseDimOrSymbolId(dims, symbols, false);
+    if (!elt.empty()) {
+      dims.push_back(elt);
+      return ParseSuccess;
+    }
+    return ParseFailure;
+  };
+
+  return parseCommaSeparatedList(Token::r_paren, parseElt);
+}
+
+/// Affine map definition.
+///
+///  affine-map-inline ::= dim-and-symbol-id-lists `->` multi-dim-affine-expr
+///                        ( `size` `(` dim-size (`,` dim-size)* `)` )?
+///  dim-size ::= affine-expr | `min` `(` affine-expr ( `,` affine-expr)+ `)`
+///
+AffineMap *Parser::parseAffineMapInline(StringRef mapId) {
+  SmallVector<StringRef, 4> dims;
+  SmallVector<StringRef, 4> symbols;
+
+  // List of dimensional identifiers.
+  if (parseDimIdList(dims, symbols)) return nullptr;
+
+  // Symbols are optional.
+  if (curToken.is(Token::l_bracket)) {
+    if (parseSymbolIdList(dims, symbols)) return nullptr;
+  }
+  if (!consumeIf(Token::arrow)) {
+    emitError("expected '->' or '['");
+    return nullptr;
+  }
+  if (!consumeIf(Token::l_paren)) {
+    emitError("expected '(' at start of affine map range");
+    return nullptr;
+  }
+
+  AffineMapParserState affState(dims, symbols);
+
+  SmallVector<AffineExpr *, 4> exprs;
+  auto parseElt = [&]() -> ParseResult {
+    auto elt = parseAffineExpr(affState);
+    ParseResult res = elt ? ParseSuccess : ParseFailure;
+    exprs.push_back(elt);
+    return res;
+  };
+
+  // Parse a multi-dimensional affine expression (a comma-separated list of 1-d
+  // affine expressions)
+  if (parseCommaSeparatedList(Token::r_paren, parseElt, false)) return nullptr;
+
+  // Parsed a valid affine map
+  auto *affineMap =
+      AffineMap::get(affState.dimCount(), affState.symbolCount(), exprs,
+                     context);
+
+  return affineMap;
 }
 
 //===----------------------------------------------------------------------===//
@@ -525,8 +754,8 @@ ParseResult Parser::parseFunctionSignature(StringRef &name,
   if (curToken.isNot(Token::l_paren))
     return emitError("expected '(' in function signature");
 
-   SmallVector<Type*, 4> arguments;
-   if (parseTypeList(arguments))
+  SmallVector<Type*, 4> arguments;
+  if (parseTypeList(arguments))
     return ParseFailure;
 
   // Parse the return type if present.
@@ -563,7 +792,7 @@ namespace {
 /// function as we are parsing it, e.g. the names for basic blocks.  It handles
 /// forward references.
 class CFGFunctionParserState {
-public:
+ public:
   CFGFunction *function;
   llvm::StringMap<std::pair<BasicBlock*, SMLoc>> blocksByName;
 
@@ -851,3 +1080,4 @@ Module *mlir::parseSourceFile(llvm::SourceMgr &sourceMgr, MLIRContext *context,
                               const SMDiagnosticHandlerTy &errorReporter) {
   return Parser(sourceMgr, context, errorReporter).parseModule();
 }
+
diff --git a/mlir/lib/Parser/Token.cpp b/mlir/lib/Parser/Token.cpp
index ca88b06ac41b..5563255b4444 100644
--- a/mlir/lib/Parser/Token.cpp
+++ b/mlir/lib/Parser/Token.cpp
@@ -65,6 +65,7 @@ StringRef Token::getTokenSpelling(Kind kind) {
    switch (kind) {
    default: assert(0 && "This token kind has no fixed spelling");
 #define TOK_PUNCTUATION(NAME, SPELLING) case NAME: return SPELLING;
+#define TOK_OPERATOR(NAME, SPELLING) case NAME: return SPELLING;
 #define TOK_KEYWORD(SPELLING) case kw_##SPELLING: return #SPELLING;
 #include "TokenKinds.def"
    }
diff --git a/mlir/lib/Parser/Token.h b/mlir/lib/Parser/Token.h
index 9c4d4f96def9..e5e4fc41886e 100644
--- a/mlir/lib/Parser/Token.h
+++ b/mlir/lib/Parser/Token.h
@@ -32,6 +32,7 @@ public:
 #define TOK_IDENTIFIER(NAME) NAME,
 #define TOK_LITERAL(NAME) NAME,
 #define TOK_PUNCTUATION(NAME, SPELLING) NAME,
+#define TOK_OPERATOR(NAME, SPELLING) NAME,
 #define TOK_KEYWORD(SPELLING) kw_##SPELLING,
 #include "TokenKinds.def"
   };
@@ -99,3 +100,4 @@ private:
 } // end namespace mlir
 
 #endif  // MLIR_LIB_PARSER_TOKEN_H
+
diff --git a/mlir/lib/Parser/TokenKinds.def b/mlir/lib/Parser/TokenKinds.def
index 7eae4708792e..72d769a90031 100644
--- a/mlir/lib/Parser/TokenKinds.def
+++ b/mlir/lib/Parser/TokenKinds.def
@@ -21,7 +21,7 @@
 //===----------------------------------------------------------------------===//
 
 #if !defined(TOK_MARKER) && !defined(TOK_IDENTIFIER) && !defined(TOK_LITERAL)&&\
-    !defined(TOK_PUNCTUATION) && !defined(TOK_KEYWORD)
+    !defined(TOK_PUNCTUATION) && !defined(TOK_OPERATOR) && !defined(TOK_KEYWORD)
 #  error Must define one of the TOK_ macros.
 #endif
 
@@ -37,6 +37,9 @@
 #ifndef TOK_PUNCTUATION
 #define TOK_PUNCTUATION(NAME, SPELLING)
 #endif
+#ifndef TOK_OPERATOR
+#define TOK_OPERATOR(NAME, SPELLING)
+#endif
 #ifndef TOK_KEYWORD
 #define TOK_KEYWORD(SPELLING)
 #endif
@@ -66,10 +69,20 @@ TOK_PUNCTUATION(l_paren,          "(")
 TOK_PUNCTUATION(r_paren,          ")")
 TOK_PUNCTUATION(l_brace,          "{")
 TOK_PUNCTUATION(r_brace,          "}")
+TOK_PUNCTUATION(l_bracket,        "[")
+TOK_PUNCTUATION(r_bracket,        "]")
 TOK_PUNCTUATION(less,             "<")
 TOK_PUNCTUATION(greater,          ">")
+TOK_PUNCTUATION(equal,            "=")
 // TODO: More punctuation.
 
+// Operators.
+TOK_OPERATOR(plus,               "+")
+TOK_OPERATOR(star,               "*")
+TOK_OPERATOR(ceildiv,            "ceildiv")
+TOK_OPERATOR(floordiv,           "floordiv")
+// TODO: More operator tokens
+
 // Keywords.  These turn "foo" into Token::kw_foo enums.
 TOK_KEYWORD(bf16)
 TOK_KEYWORD(br)
@@ -94,4 +107,5 @@ TOK_KEYWORD(vector)
 #undef TOK_IDENTIFIER
 #undef TOK_LITERAL
 #undef TOK_PUNCTUATION
+#undef TOK_OPERATOR
 #undef TOK_KEYWORD
diff --git a/mlir/test/IR/parser-affine-map.mlir b/mlir/test/IR/parser-affine-map.mlir
new file mode 100644
index 000000000000..50dd9e2aa8d8
--- /dev/null
+++ b/mlir/test/IR/parser-affine-map.mlir
@@ -0,0 +1,7 @@
+#hello_world0 = (i, j) [s0] -> (i, j)
+#hello_world1 = (i, j) -> (i, j)
+#hello_world2 = () -> (0)
+#hello_world3 = (i, j) [s0] -> (i + s0, j)
+#hello_world4 = (i, j) [s0] -> (i + s0, j + 5)
+#hello_world5 (i, j) [s0] -> i + s0, j)
+#hello_world5 = (i, j) [s0] -> i + s0, j)