diff --git a/packages/babylon/src/tokenizer/index.js b/packages/babylon/src/tokenizer/index.js
index fa484ed017..26d003094f 100644
--- a/packages/babylon/src/tokenizer/index.js
+++ b/packages/babylon/src/tokenizer/index.js
@@ -437,6 +437,7 @@ export default class Tokenizer extends LocationParser {
     let type = code === 42 ? tt.star : tt.modulo;
     let width = 1;
     let next = this.input.charCodeAt(this.state.pos + 1);
+    const exprAllowed = this.state.exprAllowed;
 
     // Exponentiation operator **
     if (code === 42 && next === 42) {
@@ -445,7 +446,7 @@ export default class Tokenizer extends LocationParser {
       type = tt.exponent;
     }
 
-    if (next === 61) {
+    if (next === 61 && !exprAllowed) {
       width++;
       type = tt.assign;
     }
diff --git a/packages/babylon/test/fixtures/flow/anonymous-function-no-parens-types/good_13/actual.js b/packages/babylon/test/fixtures/flow/anonymous-function-no-parens-types/good_13/actual.js
new file mode 100644
index 0000000000..feb53f2b4b
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/anonymous-function-no-parens-types/good_13/actual.js
@@ -0,0 +1 @@
+const x = ():*=>{}
diff --git a/packages/babylon/test/fixtures/flow/anonymous-function-no-parens-types/good_13/expected.json b/packages/babylon/test/fixtures/flow/anonymous-function-no-parens-types/good_13/expected.json
new file mode 100644
index 0000000000..ad9ad5c630
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/anonymous-function-no-parens-types/good_13/expected.json
@@ -0,0 +1,152 @@
+{
+  "type": "File",
+  "start": 0,
+  "end": 18,
+  "loc": {
+    "start": {
+      "line": 1,
+      "column": 0
+    },
+    "end": {
+      "line": 1,
+      "column": 18
+    }
+  },
+  "program": {
+    "type": "Program",
+    "start": 0,
+    "end": 18,
+    "loc": {
+      "start": {
+        "line": 1,
+        "column": 0
+      },
+      "end": {
+        "line": 1,
+        "column": 18
+      }
+    },
+    "sourceType": "module",
+    "body": [
+      {
+        "type": "VariableDeclaration",
+        "start": 0,
+        "end": 18,
+        "loc": {
+          "start": {
+            "line": 1,
+            "column": 0
+          },
+          "end": {
+            "line": 1,
+            "column": 18
+          }
+        },
+        "declarations": [
+          {
+            "type": "VariableDeclarator",
+            "start": 6,
+            "end": 18,
+            "loc": {
+              "start": {
+                "line": 1,
+                "column": 6
+              },
+              "end": {
+                "line": 1,
+                "column": 18
+              }
+            },
+            "id": {
+              "type": "Identifier",
+              "start": 6,
+              "end": 7,
+              "loc": {
+                "start": {
+                  "line": 1,
+                  "column": 6
+                },
+                "end": {
+                  "line": 1,
+                  "column": 7
+                },
+                "identifierName": "x"
+              },
+              "name": "x"
+            },
+            "init": {
+              "type": "ArrowFunctionExpression",
+              "start": 10,
+              "end": 18,
+              "loc": {
+                "start": {
+                  "line": 1,
+                  "column": 10
+                },
+                "end": {
+                  "line": 1,
+                  "column": 18
+                }
+              },
+              "predicate": null,
+              "returnType": {
+                "type": "TypeAnnotation",
+                "start": 12,
+                "end": 14,
+                "loc": {
+                  "start": {
+                    "line": 1,
+                    "column": 12
+                  },
+                  "end": {
+                    "line": 1,
+                    "column": 14
+                  }
+                },
+                "typeAnnotation": {
+                  "type": "ExistsTypeAnnotation",
+                  "start": 13,
+                  "end": 14,
+                  "loc": {
+                    "start": {
+                      "line": 1,
+                      "column": 13
+                    },
+                    "end": {
+                      "line": 1,
+                      "column": 14
+                    }
+                  }
+                }
+              },
+              "id": null,
+              "generator": false,
+              "expression": false,
+              "async": false,
+              "params": [],
+              "body": {
+                "type": "BlockStatement",
+                "start": 16,
+                "end": 18,
+                "loc": {
+                  "start": {
+                    "line": 1,
+                    "column": 16
+                  },
+                  "end": {
+                    "line": 1,
+                    "column": 18
+                  }
+                },
+                "body": [],
+                "directives": []
+              }
+            }
+          }
+        ],
+        "kind": "const"
+      }
+    ],
+    "directives": []
+  }
+}
diff --git a/packages/babylon/test/fixtures/flow/classes/good_01/actual.js b/packages/babylon/test/fixtures/flow/classes/good_01/actual.js
new file mode 100644
index 0000000000..aabb328bdf
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/classes/good_01/actual.js
@@ -0,0 +1 @@
+class C { field:*=null }
diff --git a/packages/babylon/test/fixtures/flow/classes/good_01/expected.json b/packages/babylon/test/fixtures/flow/classes/good_01/expected.json
new file mode 100644
index 0000000000..85cf0ca075
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/classes/good_01/expected.json
@@ -0,0 +1,164 @@
+{
+  "type": "File",
+  "start": 0,
+  "end": 24,
+  "loc": {
+    "start": {
+      "line": 1,
+      "column": 0
+    },
+    "end": {
+      "line": 1,
+      "column": 24
+    }
+  },
+  "program": {
+    "type": "Program",
+    "start": 0,
+    "end": 24,
+    "loc": {
+      "start": {
+        "line": 1,
+        "column": 0
+      },
+      "end": {
+        "line": 1,
+        "column": 24
+      }
+    },
+    "sourceType": "module",
+    "body": [
+      {
+        "type": "ClassDeclaration",
+        "start": 0,
+        "end": 24,
+        "loc": {
+          "start": {
+            "line": 1,
+            "column": 0
+          },
+          "end": {
+            "line": 1,
+            "column": 24
+          }
+        },
+        "id": {
+          "type": "Identifier",
+          "start": 6,
+          "end": 7,
+          "loc": {
+            "start": {
+              "line": 1,
+              "column": 6
+            },
+            "end": {
+              "line": 1,
+              "column": 7
+            },
+            "identifierName": "C"
+          },
+          "name": "C"
+        },
+        "superClass": null,
+        "body": {
+          "type": "ClassBody",
+          "start": 8,
+          "end": 24,
+          "loc": {
+            "start": {
+              "line": 1,
+              "column": 8
+            },
+            "end": {
+              "line": 1,
+              "column": 24
+            }
+          },
+          "body": [
+            {
+              "type": "ClassProperty",
+              "start": 10,
+              "end": 22,
+              "loc": {
+                "start": {
+                  "line": 1,
+                  "column": 10
+                },
+                "end": {
+                  "line": 1,
+                  "column": 22
+                }
+              },
+              "static": false,
+              "key": {
+                "type": "Identifier",
+                "start": 10,
+                "end": 15,
+                "loc": {
+                  "start": {
+                    "line": 1,
+                    "column": 10
+                  },
+                  "end": {
+                    "line": 1,
+                    "column": 15
+                  },
+                  "identifierName": "field"
+                },
+                "name": "field"
+              },
+              "computed": false,
+              "variance": null,
+              "typeAnnotation": {
+                "type": "TypeAnnotation",
+                "start": 15,
+                "end": 17,
+                "loc": {
+                  "start": {
+                    "line": 1,
+                    "column": 15
+                  },
+                  "end": {
+                    "line": 1,
+                    "column": 17
+                  }
+                },
+                "typeAnnotation": {
+                  "type": "ExistsTypeAnnotation",
+                  "start": 16,
+                  "end": 17,
+                  "loc": {
+                    "start": {
+                      "line": 1,
+                      "column": 16
+                    },
+                    "end": {
+                      "line": 1,
+                      "column": 17
+                    }
+                  }
+                }
+              },
+              "value": {
+                "type": "NullLiteral",
+                "start": 18,
+                "end": 22,
+                "loc": {
+                  "start": {
+                    "line": 1,
+                    "column": 18
+                  },
+                  "end": {
+                    "line": 1,
+                    "column": 22
+                  }
+                }
+              }
+            }
+          ]
+        }
+      }
+    ],
+    "directives": []
+  }
+}
\ No newline at end of file
diff --git a/packages/babylon/test/fixtures/flow/classes/good_01/options.json b/packages/babylon/test/fixtures/flow/classes/good_01/options.json
new file mode 100644
index 0000000000..65c79d81c7
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/classes/good_01/options.json
@@ -0,0 +1,3 @@
+{
+  "plugins": ["flow", "classProperties"]
+}