diff --git a/Doc/library/datetime.rst b/Doc/library/datetime.rst
index 8d91f4ef934..1ac2570eae5 100644
--- a/Doc/library/datetime.rst
+++ b/Doc/library/datetime.rst
@@ -1058,8 +1058,7 @@ Instance methods:
 
    If provided, *tz* must be an instance of a :class:`tzinfo` subclass, and its
    :meth:`utcoffset` and :meth:`dst` methods must not return ``None``.  If *self*
-   is naive (``self.tzinfo is None``), it is presumed to represent time in the
-   system timezone.
+   is naive, it is presumed to represent time in the system timezone.
 
    If called without arguments (or with ``tz=None``) the system local
    timezone is assumed for the target timezone.  The ``.tzinfo`` attribute of the converted
diff --git a/Lib/datetime.py b/Lib/datetime.py
index 5e9aab97002..5e922c80b01 100644
--- a/Lib/datetime.py
+++ b/Lib/datetime.py
@@ -1773,14 +1773,17 @@ class datetime(date):
         mytz = self.tzinfo
         if mytz is None:
             mytz = self._local_timezone()
+            myoffset = mytz.utcoffset(self)
+        else:
+            myoffset = mytz.utcoffset(self)
+            if myoffset is None:
+                mytz = self.replace(tzinfo=None)._local_timezone()
+                myoffset = mytz.utcoffset(self)
 
         if tz is mytz:
             return self
 
         # Convert self to UTC, and attach the new time zone object.
-        myoffset = mytz.utcoffset(self)
-        if myoffset is None:
-            raise ValueError("astimezone() requires an aware datetime")
         utc = (self - myoffset).replace(tzinfo=tz)
 
         # Convert from UTC to tz's local time.
diff --git a/Lib/test/datetimetester.py b/Lib/test/datetimetester.py
index a7e5e0b4244..7d4cdac9f41 100644
--- a/Lib/test/datetimetester.py
+++ b/Lib/test/datetimetester.py
@@ -2414,25 +2414,24 @@ class TestDateTime(TestDate):
         base = cls(2000, 2, 29)
         self.assertRaises(ValueError, base.replace, year=2001)
 
+    @support.run_with_tz('EDT4')
     def test_astimezone(self):
-        return  # The rest is no longer applicable
-        # Pretty boring!  The TZ test is more interesting here.  astimezone()
-        # simply can't be applied to a naive object.
         dt = self.theclass.now()
-        f = FixedOffset(44, "")
-        self.assertRaises(ValueError, dt.astimezone) # naive
+        f = FixedOffset(44, "0044")
+        dt_utc = dt.replace(tzinfo=timezone(timedelta(hours=-4), 'EDT'))
+        self.assertEqual(dt.astimezone(), dt_utc) # naive
         self.assertRaises(TypeError, dt.astimezone, f, f) # too many args
         self.assertRaises(TypeError, dt.astimezone, dt) # arg wrong type
-        self.assertRaises(ValueError, dt.astimezone, f) # naive
-        self.assertRaises(ValueError, dt.astimezone, tz=f)  # naive
+        dt_f = dt.replace(tzinfo=f) + timedelta(hours=4, minutes=44)
+        self.assertEqual(dt.astimezone(f), dt_f) # naive
+        self.assertEqual(dt.astimezone(tz=f), dt_f) # naive
 
         class Bogus(tzinfo):
             def utcoffset(self, dt): return None
             def dst(self, dt): return timedelta(0)
         bog = Bogus()
         self.assertRaises(ValueError, dt.astimezone, bog)   # naive
-        self.assertRaises(ValueError,
-                          dt.replace(tzinfo=bog).astimezone, f)
+        self.assertEqual(dt.replace(tzinfo=bog).astimezone(f), dt_f)
 
         class AlsoBogus(tzinfo):
             def utcoffset(self, dt): return timedelta(0)
@@ -2440,6 +2439,14 @@ class TestDateTime(TestDate):
         alsobog = AlsoBogus()
         self.assertRaises(ValueError, dt.astimezone, alsobog) # also naive
 
+        class Broken(tzinfo):
+            def utcoffset(self, dt): return 1
+            def dst(self, dt): return 1
+        broken = Broken()
+        dt_broken = dt.replace(tzinfo=broken)
+        with self.assertRaises(TypeError):
+            dt_broken.astimezone()
+
     def test_subclass_datetime(self):
 
         class C(self.theclass):
diff --git a/Misc/NEWS.d/next/Library/2018-06-10-13-26-02.bpo-33812.frGAOr.rst b/Misc/NEWS.d/next/Library/2018-06-10-13-26-02.bpo-33812.frGAOr.rst
new file mode 100644
index 00000000000..0dc3df6a795
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2018-06-10-13-26-02.bpo-33812.frGAOr.rst
@@ -0,0 +1,2 @@
+Datetime instance d with non-None tzinfo, but with d.tzinfo.utcoffset(d)
+returning None is now treated as naive by the astimezone() method.
diff --git a/Modules/_datetimemodule.c b/Modules/_datetimemodule.c
index cc7eee6fd7f..31aa88d4a26 100644
--- a/Modules/_datetimemodule.c
+++ b/Modules/_datetimemodule.c
@@ -5576,6 +5576,7 @@ datetime_astimezone(PyDateTime_DateTime *self, PyObject *args, PyObject *kw)
         return NULL;
 
     if (!HASTZINFO(self) || self->tzinfo == Py_None) {
+  naive:
         self_tzinfo = local_timezone_from_local(self);
         if (self_tzinfo == NULL)
             return NULL;
@@ -5596,6 +5597,16 @@ datetime_astimezone(PyDateTime_DateTime *self, PyObject *args, PyObject *kw)
     Py_DECREF(self_tzinfo);
     if (offset == NULL)
         return NULL;
+    else if(offset == Py_None) {
+        Py_DECREF(offset);
+        goto naive;
+    }
+    else if (!PyDelta_Check(offset)) {
+        Py_DECREF(offset);
+        PyErr_Format(PyExc_TypeError, "utcoffset() returned %.200s,"
+                     " expected timedelta or None", Py_TYPE(offset)->tp_name);
+        return NULL;
+    }
     /* result = self - offset */
     result = (PyDateTime_DateTime *)add_datetime_timedelta(self,
                                        (PyDateTime_Delta *)offset, -1);