diff --git a/CHANGES.md b/CHANGES.md
index 11341779f58..7352b857075 100644
--- a/CHANGES.md
+++ b/CHANGES.md
@@ -1,5 +1,12 @@
 ## Change Log
 
+### Unreleased
+
+#### _Black_
+
+- `Black` now respects `--skip-string-normalization` when normalizing multiline
+  docstring quotes (#1637)
+
 ### 20.8b1
 
 #### _Packaging_
diff --git a/docs/change_log.md b/docs/change_log.md
index 658414bf967..b7337166659 100644
--- a/docs/change_log.md
+++ b/docs/change_log.md
@@ -2,6 +2,13 @@
 
 ## Change Log
 
+### Unreleased
+
+#### _Black_
+
+- `Black` now respects `--skip-string-normalization` when normalizing multiline
+  docstring quotes (#1637)
+
 ### 20.8b1
 
 #### _Packaging_
diff --git a/src/black/__init__.py b/src/black/__init__.py
index 200e31fd458..f325dfc92ad 100644
--- a/src/black/__init__.py
+++ b/src/black/__init__.py
@@ -2050,7 +2050,6 @@ def visit_STRING(self, leaf: Leaf) -> Iterator[Line]:
                 if leaf.value[tail_len + 1] == docstring[-1]:
                     docstring = docstring + " "
             leaf.value = leaf.value[0:lead_len] + docstring + leaf.value[tail_len:]
-            normalize_string_quotes(leaf)
 
         yield from self.visit_default(leaf)
 
diff --git a/tests/data/docstring_no_string_normalization.py b/tests/data/docstring_no_string_normalization.py
new file mode 100644
index 00000000000..0457fcf114f
--- /dev/null
+++ b/tests/data/docstring_no_string_normalization.py
@@ -0,0 +1,209 @@
+class ALonelyClass:
+    '''
+    A multiline class docstring.
+    '''
+    def AnEquallyLonelyMethod(self):
+        '''
+        A multiline method docstring'''
+        pass
+
+
+def one_function():
+    '''This is a docstring with a single line of text.'''
+    pass
+
+
+def shockingly_the_quotes_are_normalized():
+    '''This is a multiline docstring.
+    This is a multiline docstring.
+    This is a multiline docstring.
+    '''
+    pass
+
+
+def foo():
+  """This is a docstring with             
+  some lines of text here
+  """
+  return
+
+
+def baz():
+  '''"This" is a string with some
+  embedded "quotes"'''
+  return
+
+
+def poit():
+  """
+  Lorem ipsum dolor sit amet.       
+
+  Consectetur adipiscing elit:
+   - sed do eiusmod tempor incididunt ut labore
+   - dolore magna aliqua
+     - enim ad minim veniam
+     - quis nostrud exercitation ullamco laboris nisi
+   - aliquip ex ea commodo consequat
+  """
+  pass
+
+
+def under_indent():
+  """
+  These lines are indented in a way that does not
+make sense.
+  """
+  pass
+
+
+def over_indent():
+  """
+  This has a shallow indent
+    - But some lines are deeper
+    - And the closing quote is too deep
+    """
+  pass
+
+
+def single_line():
+    """But with a newline after it!
+
+    """
+    pass
+
+
+def this():
+    r"""
+    'hey ho'
+    """
+
+
+def that():
+  """ "hey yah" """
+
+
+def and_that():
+  """
+  "hey yah" """
+
+
+def and_this():
+  ''' 
+  "hey yah"'''
+
+
+def believe_it_or_not_this_is_in_the_py_stdlib(): ''' 
+"hey yah"'''
+
+
+def shockingly_the_quotes_are_normalized_v2():
+    '''
+    Docstring Docstring Docstring
+    '''
+    pass
+
+# output
+
+class ALonelyClass:
+    '''
+    A multiline class docstring.
+    '''
+
+    def AnEquallyLonelyMethod(self):
+        '''
+        A multiline method docstring'''
+        pass
+
+
+def one_function():
+    '''This is a docstring with a single line of text.'''
+    pass
+
+
+def shockingly_the_quotes_are_normalized():
+    '''This is a multiline docstring.
+    This is a multiline docstring.
+    This is a multiline docstring.
+    '''
+    pass
+
+
+def foo():
+    """This is a docstring with
+    some lines of text here
+    """
+    return
+
+
+def baz():
+    '''"This" is a string with some
+    embedded "quotes"'''
+    return
+
+
+def poit():
+    """
+    Lorem ipsum dolor sit amet.
+
+    Consectetur adipiscing elit:
+     - sed do eiusmod tempor incididunt ut labore
+     - dolore magna aliqua
+       - enim ad minim veniam
+       - quis nostrud exercitation ullamco laboris nisi
+     - aliquip ex ea commodo consequat
+    """
+    pass
+
+
+def under_indent():
+    """
+      These lines are indented in a way that does not
+    make sense.
+    """
+    pass
+
+
+def over_indent():
+    """
+    This has a shallow indent
+      - But some lines are deeper
+      - And the closing quote is too deep
+    """
+    pass
+
+
+def single_line():
+    """But with a newline after it!"""
+    pass
+
+
+def this():
+    r"""
+    'hey ho'
+    """
+
+
+def that():
+    """ "hey yah" """
+
+
+def and_that():
+    """
+    "hey yah" """
+
+
+def and_this():
+    '''
+    "hey yah"'''
+
+
+def believe_it_or_not_this_is_in_the_py_stdlib():
+    '''
+    "hey yah"'''
+
+
+def shockingly_the_quotes_are_normalized_v2():
+    '''
+    Docstring Docstring Docstring
+    '''
+    pass
diff --git a/tests/test_black.py b/tests/test_black.py
index f5d4e1115a8..629afc5b0ad 100644
--- a/tests/test_black.py
+++ b/tests/test_black.py
@@ -517,11 +517,16 @@ def test_docstring(self) -> None:
         self.assertFormatEqual(expected, actual)
         black.assert_equivalent(source, actual)
         black.assert_stable(source, actual, DEFAULT_MODE)
+
+    @patch("black.dump_to_file", dump_to_stderr)
+    def test_docstring_no_string_normalization(self) -> None:
+        """Like test_docstring but with string normalization off."""
+        source, expected = read_data("docstring_no_string_normalization")
         mode = replace(DEFAULT_MODE, string_normalization=False)
-        not_normalized = fs(source, mode=mode)
-        self.assertFormatEqual(expected, not_normalized)
-        black.assert_equivalent(source, not_normalized)
-        black.assert_stable(source, not_normalized, mode=mode)
+        actual = fs(source, mode=mode)
+        self.assertFormatEqual(expected, actual)
+        black.assert_equivalent(source, actual)
+        black.assert_stable(source, actual, mode)
 
     def test_long_strings(self) -> None:
         """Tests for splitting long strings."""