diff --git a/mlir/include/mlir/Dialect/Shape/IR/ShapeOps.td b/mlir/include/mlir/Dialect/Shape/IR/ShapeOps.td
--- a/mlir/include/mlir/Dialect/Shape/IR/ShapeOps.td
+++ b/mlir/include/mlir/Dialect/Shape/IR/ShapeOps.td
@@ -131,6 +131,8 @@
   let results = (outs Shape_SizeType:$result);
 
   let assemblyFormat = "attr-dict $value";
+
+  let hasFolder = 1;
 }
 
 def Shape_FromExtentsOp : Shape_Op<"from_extents", [
@@ -190,6 +192,24 @@
   let hasFolder = 1;
 }
 
+def Shape_IndexToSizeOp : Shape_Op<"index_to_size", [
+    NoSideEffect,
+    DeclareOpInterfaceMethods<InferTypeOpInterface>]> {
+  let summary = "Converts a standard index to a shape size";
+  let description = [{
+    Converts a standard index to a `shape.size`.
+    This operation and its inverse, `size_to_index`, facilitate index conversion
+    between the standard and the shape dialect.
+  }];
+
+  let arguments = (ins Index:$arg);
+  let results = (outs Shape_SizeType:$result);
+
+  let assemblyFormat = "attr-dict $arg";
+
+  let hasFolder = 1;
+}
+
 def Shape_JoinOp : Shape_Op<"join", []> {
   let summary = "Returns the least general shape.size of its operands";
   let description = [{
@@ -288,6 +308,24 @@
   let hasFolder = 1;
 }
 
+def Shape_SizeToIndexOp : Shape_Op<"size_to_index", [
+    NoSideEffect,
+    DeclareOpInterfaceMethods<InferTypeOpInterface>]> {
+  let summary = "Casts between index types of the shape and standard dialect";
+  let description = [{
+    Converts a `shape.size` to a standard index.
+    This operation and its inverse, `index_to_size`, facilitate index conversion
+    between the standard and the shape dialect.
+  }];
+
+  let arguments = (ins Shape_SizeType:$arg);
+  let results = (outs Index:$result);
+
+  let assemblyFormat = "attr-dict $arg";
+
+  let hasFolder = 1;
+}
+
 def Shape_YieldOp : Shape_Op<"yield", [NoSideEffect, Terminator]> {
   let summary = "Returns the value to parent op";
 
@@ -499,7 +537,6 @@
   let assemblyFormat = "$inputs attr-dict";
 }
 
-
 // Canonicalization patterns.
 
 #endif // SHAPE_OPS
diff --git a/mlir/lib/Dialect/Shape/IR/Shape.cpp b/mlir/lib/Dialect/Shape/IR/Shape.cpp
--- a/mlir/lib/Dialect/Shape/IR/Shape.cpp
+++ b/mlir/lib/Dialect/Shape/IR/Shape.cpp
@@ -249,7 +249,7 @@
   return success();
 }
 
-OpFoldResult ConstShapeOp::fold(ArrayRef<Attribute>) { return shape(); }
+OpFoldResult ConstShapeOp::fold(ArrayRef<Attribute>) { return shapeAttr(); }
 
 LogicalResult
 ConstShapeOp::inferReturnTypes(MLIRContext *context,
@@ -273,6 +273,29 @@
   return success();
 }
 
+OpFoldResult ConstSizeOp::fold(ArrayRef<Attribute>) { return valueAttr(); }
+
+//===----------------------------------------------------------------------===//
+// IndexToSizeOp
+//===----------------------------------------------------------------------===//
+
+OpFoldResult IndexToSizeOp::fold(ArrayRef<Attribute> operands) {
+
+  // Constant values of both types, `shape.size` and `index`, are represented as
+  // `IntegerAttr`s which makes constant folding simple.
+  if (Attribute arg = operands[0])
+    return arg;
+  return {};
+}
+
+LogicalResult IndexToSizeOp::inferReturnTypes(
+    MLIRContext *context, Optional<Location> location, ValueRange operands,
+    DictionaryAttr attributes, RegionRange regions,
+    SmallVectorImpl<Type> &inferredReturnTypes) {
+  inferredReturnTypes.push_back(SizeType::get(context));
+  return success();
+}
+
 //===----------------------------------------------------------------------===//
 // FromExtentsOp
 //===----------------------------------------------------------------------===//
@@ -316,6 +339,27 @@
   return builder.getI64TensorAttr(type.getShape());
 }
 
+//===----------------------------------------------------------------------===//
+// SizeToIndexOp
+//===----------------------------------------------------------------------===//
+
+OpFoldResult SizeToIndexOp::fold(ArrayRef<Attribute> operands) {
+
+  // Constant values of both types, `shape.size` and `index`, are represented as
+  // `IntegerAttr`s which makes constant folding simple.
+  if (Attribute arg = operands[0])
+    return arg;
+  return {};
+}
+
+LogicalResult SizeToIndexOp::inferReturnTypes(
+    MLIRContext *context, Optional<Location> location, ValueRange operands,
+    DictionaryAttr attributes, RegionRange regions,
+    SmallVectorImpl<Type> &inferredReturnTypes) {
+  inferredReturnTypes.push_back(IndexType::get(context));
+  return success();
+}
+
 //===----------------------------------------------------------------------===//
 // SplitAtOp
 //===----------------------------------------------------------------------===//
diff --git a/mlir/test/Dialect/Shape/canonicalize.mlir b/mlir/test/Dialect/Shape/canonicalize.mlir
--- a/mlir/test/Dialect/Shape/canonicalize.mlir
+++ b/mlir/test/Dialect/Shape/canonicalize.mlir
@@ -88,6 +88,7 @@
 }
 
 // -----
+
 // Basic case.
 // CHECK-LABEL: func @f()
 func @f() -> !shape.shape {
@@ -106,3 +107,56 @@
   %ret = shape.from_extents %e0, %arg0
   return %ret : !shape.shape
 }
+
+// -----
+// Cast constant size to index and fold it away.
+// CHECK-LABEL: func @const_size_to_index
+func @const_size_to_index() -> index {
+  // CHECK-NOT: shape.index_cast
+  %cs = shape.const_size 123
+  // CHECK: constant 123 : index
+  %ci = shape.size_to_index %cs
+  return %ci : index
+}
+
+// -----
+// Cast constant index to size and fold it away.
+// CHECK-LABEL: func @const_index_to_size
+func @const_index_to_size() -> !shape.size {
+  // CHECK-NOT: index_cast
+  %ci = constant 123 : index
+  // CHECK: shape.const_size 123
+  %cs = shape.index_to_size %ci
+  return %cs : !shape.size
+}
+
+// -----
+// Cast constant index to size, then back, and fold it away.
+// CHECK-LABEL: func @const_index_to_size_to_index
+func @const_index_to_size_to_index() -> index {
+  // CHECK-NOT: shape.index_cast
+  %ci0 = constant 123 : index
+  %cs0 = shape.index_to_size %ci0
+  // CHECK: %[[CI:.*]] = constant 123 : index
+  // CHECK-NEXT: return %[[CI]] : index
+  %ci1 = shape.size_to_index %cs0
+  return %ci1 : index
+}
+
+// -----
+// No folding.
+// CHECK-LABEL: func @nonfoldable_size_to_index
+func @nonfoldable_size_to_index(%cs : !shape.size) -> index {
+  // CHECK: shape.size_to_index
+  %ci = shape.size_to_index %cs
+  return %ci : index
+}
+
+// -----
+// No folding.
+// CHECK-LABEL: func @nonfoldable_index_to_size
+func @nonfoldable_index_to_size(%ci : index) -> !shape.size {
+  // CHECK: shape.index_to_size
+  %cs = shape.index_to_size %ci
+  return %cs : !shape.size
+}