diff --git a/doc/api/stream.md b/doc/api/stream.md
index cb6d23e52583ec..07755e05dc2f65 100644
--- a/doc/api/stream.md
+++ b/doc/api/stream.md
@@ -281,6 +281,9 @@ The stream is not closed when the `'error'` event is emitted unless the
 [`autoDestroy`][writable-new] option was set to `true` when creating the
 stream.
 
+After `'error'`, no further events other than `'close'` *should* be emitted
+(including `'error'` events).
+
 ##### Event: 'finish'
 <!-- YAML
 added: v0.9.4
diff --git a/lib/_stream_readable.js b/lib/_stream_readable.js
index 1830eea66decc9..c60d84ffb87b00 100644
--- a/lib/_stream_readable.js
+++ b/lib/_stream_readable.js
@@ -117,6 +117,9 @@ function ReadableState(options, stream, isDuplex) {
   this.resumeScheduled = false;
   this.paused = true;
 
+  // True if the error was already emitted and should not be thrown again
+  this.errorEmitted = false;
+
   // Should close be emitted on destroy. Defaults to true.
   this.emitClose = options.emitClose !== false;
 
diff --git a/lib/_stream_writable.js b/lib/_stream_writable.js
index 5d52b0d2a4a179..315360ca6b6ac7 100644
--- a/lib/_stream_writable.js
+++ b/lib/_stream_writable.js
@@ -429,13 +429,11 @@ function onwriteError(stream, state, sync, er, cb) {
     // This can emit finish, and it will always happen
     // after error
     process.nextTick(finishMaybe, stream, state);
-    stream._writableState.errorEmitted = true;
     errorOrDestroy(stream, er);
   } else {
     // The caller expect this to happen before if
     // it is async
     cb(er);
-    stream._writableState.errorEmitted = true;
     errorOrDestroy(stream, er);
     // This can emit finish, but finish must
     // always follow error
diff --git a/lib/internal/streams/destroy.js b/lib/internal/streams/destroy.js
index 200c75459ad56d..0d3eb6327b9d75 100644
--- a/lib/internal/streams/destroy.js
+++ b/lib/internal/streams/destroy.js
@@ -1,22 +1,37 @@
 'use strict';
 
+function needError(stream, err) {
+  if (!err) {
+    return false;
+  }
+
+  const r = stream._readableState;
+  const w = stream._writableState;
+
+  if ((w && w.errorEmitted) || (r && r.errorEmitted)) {
+    return false;
+  }
+
+  if (w) {
+    w.errorEmitted = true;
+  }
+  if (r) {
+    r.errorEmitted = true;
+  }
+
+  return true;
+}
+
 // Undocumented cb() API, needed for core, not for public API
 function destroy(err, cb) {
-  const readableDestroyed = this._readableState &&
-    this._readableState.destroyed;
-  const writableDestroyed = this._writableState &&
-    this._writableState.destroyed;
+  const r = this._readableState;
+  const w = this._writableState;
 
-  if (readableDestroyed || writableDestroyed) {
+  if ((w && w.destroyed) || (r && r.destroyed)) {
     if (cb) {
       cb(err);
-    } else if (err) {
-      if (!this._writableState) {
-        process.nextTick(emitErrorNT, this, err);
-      } else if (!this._writableState.errorEmitted) {
-        this._writableState.errorEmitted = true;
-        process.nextTick(emitErrorNT, this, err);
-      }
+    } else if (needError(this, err)) {
+      process.nextTick(emitErrorNT, this, err);
     }
 
     return this;
@@ -25,28 +40,19 @@ function destroy(err, cb) {
   // We set destroyed to true before firing error callbacks in order
   // to make it re-entrance safe in case destroy() is called within callbacks
 
-  if (this._readableState) {
-    this._readableState.destroyed = true;
+  if (w) {
+    w.destroyed = true;
   }
-
-  // If this is a duplex stream mark the writable part as destroyed as well
-  if (this._writableState) {
-    this._writableState.destroyed = true;
+  if (r) {
+    r.destroyed = true;
   }
 
   this._destroy(err || null, (err) => {
-    if (!cb && err) {
-      if (!this._writableState) {
-        process.nextTick(emitErrorAndCloseNT, this, err);
-      } else if (!this._writableState.errorEmitted) {
-        this._writableState.errorEmitted = true;
-        process.nextTick(emitErrorAndCloseNT, this, err);
-      } else {
-        process.nextTick(emitCloseNT, this);
-      }
-    } else if (cb) {
+    if (cb) {
       process.nextTick(emitCloseNT, this);
       cb(err);
+    } else if (needError(this, err)) {
+      process.nextTick(emitErrorAndCloseNT, this, err);
     } else {
       process.nextTick(emitCloseNT, this);
     }
@@ -61,29 +67,36 @@ function emitErrorAndCloseNT(self, err) {
 }
 
 function emitCloseNT(self) {
-  if (self._writableState && !self._writableState.emitClose)
+  const r = self._readableState;
+  const w = self._writableState;
+
+  if (w && !w.emitClose)
     return;
-  if (self._readableState && !self._readableState.emitClose)
+  if (r && !r.emitClose)
     return;
   self.emit('close');
 }
 
 function undestroy() {
-  if (this._readableState) {
-    this._readableState.destroyed = false;
-    this._readableState.reading = false;
-    this._readableState.ended = false;
-    this._readableState.endEmitted = false;
+  const r = this._readableState;
+  const w = this._writableState;
+
+  if (r) {
+    r.destroyed = false;
+    r.reading = false;
+    r.ended = false;
+    r.endEmitted = false;
+    r.errorEmitted = false;
   }
 
-  if (this._writableState) {
-    this._writableState.destroyed = false;
-    this._writableState.ended = false;
-    this._writableState.ending = false;
-    this._writableState.finalCalled = false;
-    this._writableState.prefinished = false;
-    this._writableState.finished = false;
-    this._writableState.errorEmitted = false;
+  if (w) {
+    w.destroyed = false;
+    w.ended = false;
+    w.ending = false;
+    w.finalCalled = false;
+    w.prefinished = false;
+    w.finished = false;
+    w.errorEmitted = false;
   }
 }
 
@@ -98,12 +111,12 @@ function errorOrDestroy(stream, err) {
   // the error to be emitted nextTick. In a future
   // semver major update we should change the default to this.
 
-  const rState = stream._readableState;
-  const wState = stream._writableState;
+  const r = stream._readableState;
+  const w = stream._writableState;
 
-  if ((rState && rState.autoDestroy) || (wState && wState.autoDestroy))
+  if ((r && r.autoDestroy) || (w && w.autoDestroy))
     stream.destroy(err);
-  else
+  else if (needError(stream, err))
     stream.emit('error', err);
 }
 
diff --git a/test/parallel/test-net-connect-buffer.js b/test/parallel/test-net-connect-buffer.js
index 8c95400fb6a606..41df6a06662400 100644
--- a/test/parallel/test-net-connect-buffer.js
+++ b/test/parallel/test-net-connect-buffer.js
@@ -69,12 +69,14 @@ tcp.listen(0, common.mustCall(function() {
     [],
     {}
   ].forEach((value) => {
-    common.expectsError(() => socket.write(value), {
+    // We need to check the callback since 'error' will only
+    // be emitted once per instance.
+    socket.write(value, common.expectsError({
       code: 'ERR_INVALID_ARG_TYPE',
       type: TypeError,
       message: 'The "chunk" argument must be one of type string or Buffer. ' +
                `Received type ${typeof value}`
-    });
+    }));
   });
 
   // Write a string that contains a multi-byte character sequence to test that
diff --git a/test/parallel/test-stream-error-once.js b/test/parallel/test-stream-error-once.js
new file mode 100644
index 00000000000000..71f268cfa476a1
--- /dev/null
+++ b/test/parallel/test-stream-error-once.js
@@ -0,0 +1,19 @@
+'use strict';
+const common = require('../common');
+const { Writable, Readable } = require('stream');
+
+{
+  const writable = new Writable();
+  writable.on('error', common.mustCall());
+  writable.end();
+  writable.write('h');
+  writable.write('h');
+}
+
+{
+  const readable = new Readable();
+  readable.on('error', common.mustCall());
+  readable.push(null);
+  readable.push('h');
+  readable.push('h');
+}
diff --git a/test/parallel/test-stream-readable-invalid-chunk.js b/test/parallel/test-stream-readable-invalid-chunk.js
index fcd7414bb66632..3e147c065fdf54 100644
--- a/test/parallel/test-stream-readable-invalid-chunk.js
+++ b/test/parallel/test-stream-readable-invalid-chunk.js
@@ -3,17 +3,32 @@
 const common = require('../common');
 const stream = require('stream');
 
-const readable = new stream.Readable({
-  read: () => {}
-});
-
-function checkError(fn) {
-  common.expectsError(fn, {
+function testPushArg(val) {
+  const readable = new stream.Readable({
+    read: () => {}
+  });
+  readable.on('error', common.expectsError({
     code: 'ERR_INVALID_ARG_TYPE',
     type: TypeError
+  }));
+  readable.push(val);
+}
+
+testPushArg([]);
+testPushArg({});
+testPushArg(0);
+
+function testUnshiftArg(val) {
+  const readable = new stream.Readable({
+    read: () => {}
   });
+  readable.on('error', common.expectsError({
+    code: 'ERR_INVALID_ARG_TYPE',
+    type: TypeError
+  }));
+  readable.unshift(val);
 }
 
-checkError(() => readable.push([]));
-checkError(() => readable.push({}));
-checkError(() => readable.push(0));
+testUnshiftArg([]);
+testUnshiftArg({});
+testUnshiftArg(0);
diff --git a/test/parallel/test-stream-readable-unshift.js b/test/parallel/test-stream-readable-unshift.js
index d574b7d0465b76..6eefb55d73bc88 100644
--- a/test/parallel/test-stream-readable-unshift.js
+++ b/test/parallel/test-stream-readable-unshift.js
@@ -112,23 +112,6 @@ const { Readable } = require('stream');
 
 }
 
-{
-  // Check that error is thrown for invalid chunks
-
-  const readable = new Readable({ read() {} });
-  function checkError(fn) {
-    common.expectsError(fn, {
-      code: 'ERR_INVALID_ARG_TYPE',
-      type: TypeError
-    });
-  }
-
-  checkError(() => readable.unshift([]));
-  checkError(() => readable.unshift({}));
-  checkError(() => readable.unshift(0));
-
-}
-
 {
   // Check that ObjectMode works
   const readable = new Readable({ objectMode: true, read() {} });
diff --git a/test/parallel/test-stream-unshift-read-race.js b/test/parallel/test-stream-unshift-read-race.js
index f7b633338cb810..562e10776ebd91 100644
--- a/test/parallel/test-stream-unshift-read-race.js
+++ b/test/parallel/test-stream-unshift-read-race.js
@@ -86,13 +86,7 @@ w._write = function(chunk, encoding, cb) {
 };
 
 r.on('end', common.mustCall(function() {
-  common.expectsError(function() {
-    r.unshift(Buffer.allocUnsafe(1));
-  }, {
-    code: 'ERR_STREAM_UNSHIFT_AFTER_END_EVENT',
-    type: Error,
-    message: 'stream.unshift() after end event'
-  });
+  r.unshift(Buffer.allocUnsafe(1));
   w.end();
 }));
 
diff --git a/test/parallel/test-stream2-writable.js b/test/parallel/test-stream2-writable.js
index 262606d9062a10..b20f5d3f18cc13 100644
--- a/test/parallel/test-stream2-writable.js
+++ b/test/parallel/test-stream2-writable.js
@@ -402,3 +402,42 @@ const helloWorldBuffer = Buffer.from('hello world');
   w.write(Buffer.allocUnsafe(1));
   w.end(Buffer.allocUnsafe(0));
 }
+
+{
+  // Verify that error is only emitted once when failing in _finish.
+  const w = new W();
+
+  w._final = common.mustCall(function(cb) {
+    cb(new Error('test'));
+  });
+  w.on('error', common.mustCall((err) => {
+    assert.strictEqual(w._writableState.errorEmitted, true);
+    assert.strictEqual(err.message, 'test');
+    w.on('error', common.mustNotCall());
+    w.destroy(new Error());
+  }));
+  w.end();
+}
+
+{
+  // Verify that error is only emitted once when failing in write.
+  const w = new W();
+  w.on('error', common.mustCall((err) => {
+    assert.strictEqual(w._writableState.errorEmitted, true);
+    assert.strictEqual(err.code, 'ERR_STREAM_NULL_VALUES');
+  }));
+  w.write(null);
+  w.destroy(new Error());
+}
+
+{
+  // Verify that error is only emitted once when failing in write after end.
+  const w = new W();
+  w.on('error', common.mustCall((err) => {
+    assert.strictEqual(w._writableState.errorEmitted, true);
+    assert.strictEqual(err.code, 'ERR_STREAM_WRITE_AFTER_END');
+  }));
+  w.end();
+  w.write('hello');
+  w.destroy(new Error());
+}