diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5360d62ec1..2bbbb154f1 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -20,6 +20,7 @@ See [CHANGELOG - 6to5](CHANGELOG-6to5.md) for the pre-4.0.0 version changelog.
   * Ensure that invalid identifier JSX attribute keys are quoted when transforming to calls.
   * Fix ES3 property literal plugin.
   * Fix parameters after defaults in arrow functions refering to the wrong `arguments`.
+  * Fix parser bug where arrow functions have a higher precedence than they should.
 
 ## 6.0.13
 
diff --git a/packages/babylon/src/parser/expression.js b/packages/babylon/src/parser/expression.js
index b27bf3abd2..677e78a1ed 100644
--- a/packages/babylon/src/parser/expression.js
+++ b/packages/babylon/src/parser/expression.js
@@ -221,12 +221,18 @@ pp.parseMaybeUnary = function (refShorthandDefaultPos) {
 
 pp.parseExprSubscripts = function (refShorthandDefaultPos) {
   let startPos = this.state.start, startLoc = this.state.startLoc;
+  let potentialArrowAt = this.state.potentialArrowAt;
   let expr = this.parseExprAtom(refShorthandDefaultPos);
+
+  if (expr.type === "ArrowFunctionExpression" && expr.start === potentialArrowAt) {
+    return expr;
+  }
+
   if (refShorthandDefaultPos && refShorthandDefaultPos.start) {
     return expr;
-  } else {
-    return this.parseSubscripts(expr, startPos, startLoc);
   }
+
+  return this.parseSubscripts(expr, startPos, startLoc);
 };
 
 pp.parseSubscripts = function (base, startPos, startLoc, noCalls) {
@@ -250,7 +256,7 @@ pp.parseSubscripts = function (base, startPos, startLoc, noCalls) {
       this.expect(tt.bracketR);
       base = this.finishNode(node, "MemberExpression");
     } else if (!noCalls && this.match(tt.parenL)) {
-      let possibleAsync = base.type === "Identifier" && base.name === "async" && !this.canInsertSemicolon();
+      let possibleAsync = this.state.potentialArrowAt === base.start && base.type === "Identifier" && base.name === "async" && !this.canInsertSemicolon();
       this.next();
 
       let node = this.startNodeAt(startPos, startLoc);
@@ -677,7 +683,7 @@ pp.parseObj = function (isPattern, refShorthandDefaultPos) {
     if (prop.shorthand) {
       this.addExtra(prop, "shorthand", true);
     }
-    
+
     node.properties.push(prop);
   }
 
@@ -915,6 +921,9 @@ pp.parseIdentifier = function (liberal) {
 // Parses await expression inside async function.
 
 pp.parseAwait = function (node) {
+  if (!this.state.inAsync) {
+    this.unexpected();
+  }
   if (this.isLineTerminator()) {
     this.unexpected();
   }
diff --git a/packages/babylon/src/parser/statement.js b/packages/babylon/src/parser/statement.js
index ef7649dc4b..710850b555 100644
--- a/packages/babylon/src/parser/statement.js
+++ b/packages/babylon/src/parser/statement.js
@@ -128,7 +128,8 @@ pp.parseStatement = function (declaration, topLevel) {
   // simply start parsing an expression, and afterwards, if the
   // next token is a colon and the expression was a simple
   // Identifier node, we switch to interpreting it as a label.
-  let maybeName = this.state.value, expr = this.parseExpression();
+  let maybeName = this.state.value;
+  let expr = this.parseExpression();
 
   if (starttype === tt.name && expr.type === "Identifier" && this.eat(tt.colon)) {
     return this.parseLabeledStatement(node, maybeName, expr);
diff --git a/packages/babylon/src/plugins/flow.js b/packages/babylon/src/plugins/flow.js
index 25e0ced340..994d744dfb 100644
--- a/packages/babylon/src/plugins/flow.js
+++ b/packages/babylon/src/plugins/flow.js
@@ -653,6 +653,7 @@ export default function (instance) {
 
   instance.extend("parseParenItem", function () {
     return function (node, startLoc, startPos, forceArrow?) {
+      let canBeArrow = this.state.potentialArrowAt = startPos;
       if (this.match(tt.colon)) {
         let typeCastNode = this.startNodeAt(startLoc, startPos);
         typeCastNode.expression = node;
@@ -662,7 +663,7 @@ export default function (instance) {
           this.unexpected();
         }
 
-        if (this.eat(tt.arrow)) {
+        if (canBeArrow && this.eat(tt.arrow)) {
           // ((lol): number => {});
           let func = this.parseArrowExpression(this.startNodeAt(startLoc, startPos), [node]);
           func.returnType = typeCastNode.typeAnnotation;
@@ -952,7 +953,7 @@ export default function (instance) {
       startPos = startPos || this.state.start;
       startLoc = startLoc || this.state.startLoc;
 
-      if (this.lookahead().type === tt.parenR) {
+      if (canBeArrow && this.lookahead().type === tt.parenR) {
         // let foo = (): number => {};
         this.expect(tt.parenL);
         this.expect(tt.parenR);
diff --git a/packages/babylon/test/fixtures/experimental/async-functions/expression-statement/actual.js b/packages/babylon/test/fixtures/experimental/async-functions/expression-statement/actual.js
new file mode 100644
index 0000000000..2847ccae29
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/async-functions/expression-statement/actual.js
@@ -0,0 +1 @@
+async () => {}
diff --git a/packages/babylon/test/fixtures/experimental/async-functions/expression-statement/expected.json b/packages/babylon/test/fixtures/experimental/async-functions/expression-statement/expected.json
new file mode 100644
index 0000000000..b2dfa5cf58
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/async-functions/expression-statement/expected.json
@@ -0,0 +1,86 @@
+{
+  "type": "File",
+  "start": 0,
+  "end": 14,
+  "loc": {
+    "start": {
+      "line": 1,
+      "column": 0
+    },
+    "end": {
+      "line": 1,
+      "column": 14
+    }
+  },
+  "program": {
+    "type": "Program",
+    "start": 0,
+    "end": 14,
+    "loc": {
+      "start": {
+        "line": 1,
+        "column": 0
+      },
+      "end": {
+        "line": 1,
+        "column": 14
+      }
+    },
+    "sourceType": "script",
+    "body": [
+      {
+        "type": "ExpressionStatement",
+        "start": 0,
+        "end": 14,
+        "loc": {
+          "start": {
+            "line": 1,
+            "column": 0
+          },
+          "end": {
+            "line": 1,
+            "column": 14
+          }
+        },
+        "expression": {
+          "type": "ArrowFunctionExpression",
+          "start": 0,
+          "end": 14,
+          "loc": {
+            "start": {
+              "line": 1,
+              "column": 0
+            },
+            "end": {
+              "line": 1,
+              "column": 14
+            }
+          },
+          "id": null,
+          "generator": false,
+          "expression": false,
+          "async": true,
+          "params": [],
+          "body": {
+            "type": "BlockStatement",
+            "start": 12,
+            "end": 14,
+            "loc": {
+              "start": {
+                "line": 1,
+                "column": 12
+              },
+              "end": {
+                "line": 1,
+                "column": 14
+              }
+            },
+            "body": [],
+            "directives": []
+          }
+        }
+      }
+    ],
+    "directives": []
+  }
+}
\ No newline at end of file
diff --git a/packages/babylon/test/fixtures/experimental/async-functions/no-callee/actual.js b/packages/babylon/test/fixtures/experimental/async-functions/no-callee/actual.js
new file mode 100644
index 0000000000..968eff9fb2
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/async-functions/no-callee/actual.js
@@ -0,0 +1 @@
+async () => {}()
diff --git a/packages/babylon/test/fixtures/experimental/async-functions/no-callee/options.json b/packages/babylon/test/fixtures/experimental/async-functions/no-callee/options.json
new file mode 100644
index 0000000000..d50e8469b8
--- /dev/null
+++ b/packages/babylon/test/fixtures/experimental/async-functions/no-callee/options.json
@@ -0,0 +1,3 @@
+{
+  "throws": "Unexpected token (1:14)"
+}
diff --git a/packages/babylon/test/fixtures/flow/regression/issue-2154/actual.js b/packages/babylon/test/fixtures/flow/regression/issue-2154/actual.js
new file mode 100644
index 0000000000..a71a0f34e2
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/regression/issue-2154/actual.js
@@ -0,0 +1 @@
+!()=>0
diff --git a/packages/babylon/test/fixtures/flow/regression/issue-2154/options.json b/packages/babylon/test/fixtures/flow/regression/issue-2154/options.json
new file mode 100644
index 0000000000..10bd5624f0
--- /dev/null
+++ b/packages/babylon/test/fixtures/flow/regression/issue-2154/options.json
@@ -0,0 +1,3 @@
+{
+  "throws": "Unexpected token (1:2)"
+}
diff --git a/packages/babylon/test/fixtures/flow/type-annotations/79/actual.js b/packages/babylon/test/fixtures/flow/type-annotations/79/actual.js
index 72d2f265fd..f77d0b5e7f 100644
--- a/packages/babylon/test/fixtures/flow/type-annotations/79/actual.js
+++ b/packages/babylon/test/fixtures/flow/type-annotations/79/actual.js
@@ -1 +1 @@
-var foo = bar ? (foo) : number => {} : baz;
\ No newline at end of file
+var foo = bar ? (foo) : number => {} : baz;
diff --git a/packages/babylon/test/fixtures/harmony/arrow-functions/no-callee/actual.js b/packages/babylon/test/fixtures/harmony/arrow-functions/no-callee/actual.js
new file mode 100644
index 0000000000..7737f0162c
--- /dev/null
+++ b/packages/babylon/test/fixtures/harmony/arrow-functions/no-callee/actual.js
@@ -0,0 +1 @@
+() => {}()
diff --git a/packages/babylon/test/fixtures/harmony/arrow-functions/no-callee/options.json b/packages/babylon/test/fixtures/harmony/arrow-functions/no-callee/options.json
new file mode 100644
index 0000000000..9ce9658f7d
--- /dev/null
+++ b/packages/babylon/test/fixtures/harmony/arrow-functions/no-callee/options.json
@@ -0,0 +1,3 @@
+{
+  "throws": "Unexpected token (1:8)"
+}