diff --git a/Lib/test/test_urlparse.py b/Lib/test/test_urlparse.py
index b0aed37de7dcae..5e08aa1bbad89d 100644
--- a/Lib/test/test_urlparse.py
+++ b/Lib/test/test_urlparse.py
@@ -70,7 +70,9 @@
 
 class UrlParseTestCase(unittest.TestCase):
 
-    def checkRoundtrips(self, url, parsed, split):
+    def checkRoundtrips(self, url, parsed, split, url2=None):
+        if url2 is None:
+            url2 = url
         result = urllib.parse.urlparse(url)
         self.assertEqual(result, parsed)
         t = (result.scheme, result.netloc, result.path,
@@ -78,7 +80,7 @@ def checkRoundtrips(self, url, parsed, split):
         self.assertEqual(t, parsed)
         # put it back together and it should be the same
         result2 = urllib.parse.urlunparse(result)
-        self.assertEqual(result2, url)
+        self.assertEqual(result2, url2)
         self.assertEqual(result2, result.geturl())
 
         # the result of geturl() is a fixpoint; we can always parse it
@@ -104,7 +106,7 @@ def checkRoundtrips(self, url, parsed, split):
              result.query, result.fragment)
         self.assertEqual(t, split)
         result2 = urllib.parse.urlunsplit(result)
-        self.assertEqual(result2, url)
+        self.assertEqual(result2, url2)
         self.assertEqual(result2, result.geturl())
 
         # check the fixpoint property of re-parsing the result of geturl()
@@ -142,9 +144,39 @@ def test_qs(self):
 
     def test_roundtrips(self):
         str_cases = [
+            ('path/to/file',
+             ('', '', 'path/to/file', '', '', ''),
+             ('', '', 'path/to/file', '', '')),
+            ('/path/to/file',
+             ('', '', '/path/to/file', '', '', ''),
+             ('', '', '/path/to/file', '', '')),
+            ('//path/to/file',
+             ('', 'path', '/to/file', '', '', ''),
+             ('', 'path', '/to/file', '', '')),
+            ('////path/to/file',
+             ('', '', '//path/to/file', '', '', ''),
+             ('', '', '//path/to/file', '', '')),
+            ('scheme:path/to/file',
+             ('scheme', '', 'path/to/file', '', '', ''),
+             ('scheme', '', 'path/to/file', '', '')),
+            ('scheme:/path/to/file',
+             ('scheme', '', '/path/to/file', '', '', ''),
+             ('scheme', '', '/path/to/file', '', '')),
+            ('scheme://path/to/file',
+             ('scheme', 'path', '/to/file', '', '', ''),
+             ('scheme', 'path', '/to/file', '', '')),
+            ('scheme:////path/to/file',
+             ('scheme', '', '//path/to/file', '', '', ''),
+             ('scheme', '', '//path/to/file', '', '')),
             ('file:///tmp/junk.txt',
              ('file', '', '/tmp/junk.txt', '', '', ''),
              ('file', '', '/tmp/junk.txt', '', '')),
+            ('file:////tmp/junk.txt',
+             ('file', '', '//tmp/junk.txt', '', '', ''),
+             ('file', '', '//tmp/junk.txt', '', '')),
+            ('file://///tmp/junk.txt',
+             ('file', '', '///tmp/junk.txt', '', '', ''),
+             ('file', '', '///tmp/junk.txt', '', '')),
             ('imap://mail.python.org/mbox1',
              ('imap', 'mail.python.org', '/mbox1', '', '', ''),
              ('imap', 'mail.python.org', '/mbox1', '', '')),
@@ -175,6 +207,38 @@ def _encode(t):
         for url, parsed, split in str_cases + bytes_cases:
             self.checkRoundtrips(url, parsed, split)
 
+    def test_roundtrips_normalization(self):
+        str_cases = [
+            ('///path/to/file',
+             '/path/to/file',
+             ('', '', '/path/to/file', '', '', ''),
+             ('', '', '/path/to/file', '', '')),
+            ('scheme:///path/to/file',
+             'scheme:/path/to/file',
+             ('scheme', '', '/path/to/file', '', '', ''),
+             ('scheme', '', '/path/to/file', '', '')),
+            ('file:/tmp/junk.txt',
+             'file:///tmp/junk.txt',
+             ('file', '', '/tmp/junk.txt', '', '', ''),
+             ('file', '', '/tmp/junk.txt', '', '')),
+            ('http:/tmp/junk.txt',
+             'http:///tmp/junk.txt',
+             ('http', '', '/tmp/junk.txt', '', '', ''),
+             ('http', '', '/tmp/junk.txt', '', '')),
+            ('https:/tmp/junk.txt',
+             'https:///tmp/junk.txt',
+             ('https', '', '/tmp/junk.txt', '', '', ''),
+             ('https', '', '/tmp/junk.txt', '', '')),
+        ]
+        def _encode(t):
+            return (t[0].encode('ascii'),
+                    t[1].encode('ascii'),
+                    tuple(x.encode('ascii') for x in t[2]),
+                    tuple(x.encode('ascii') for x in t[3]))
+        bytes_cases = [_encode(x) for x in str_cases]
+        for url, url2, parsed, split in str_cases + bytes_cases:
+            self.checkRoundtrips(url, parsed, split, url2)
+
     def test_http_roundtrips(self):
         # urllib.parse.urlsplit treats 'http:' as an optimized special case,
         # so we test both 'http:' and 'https:' in all the following.
diff --git a/Lib/urllib/parse.py b/Lib/urllib/parse.py
index 0ab2023843f6b5..44806e67a8fc01 100644
--- a/Lib/urllib/parse.py
+++ b/Lib/urllib/parse.py
@@ -521,7 +521,7 @@ def urlunsplit(components):
     empty query; the RFC states that these are equivalent)."""
     scheme, netloc, url, query, fragment, _coerce_result = (
                                           _coerce_args(*components))
-    if netloc or (scheme and scheme in uses_netloc and url[:2] != '//'):
+    if netloc or (scheme and scheme in uses_netloc) or url[:2] == '//':
         if url and url[:1] != '/': url = '/' + url
         url = '//' + (netloc or '') + url
     if scheme:
diff --git a/Misc/NEWS.d/next/Library/2019-08-27-01-16-50.gh-issue-67693.4NIAiy.rst b/Misc/NEWS.d/next/Library/2019-08-27-01-16-50.gh-issue-67693.4NIAiy.rst
new file mode 100644
index 00000000000000..22457df03e65c9
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2019-08-27-01-16-50.gh-issue-67693.4NIAiy.rst
@@ -0,0 +1,2 @@
+Fix :func:`urllib.parse.urlunparse` and :func:`urllib.parse.urlunsplit` for URIs with path starting with multiple slashes and no authority.
+Based on patch by Ashwin Ramaswami.