diff --git a/Lib/html/parser.py b/Lib/html/parser.py
index dd9c2e14862..5c4a7ef7087 100644
--- a/Lib/html/parser.py
+++ b/Lib/html/parser.py
@@ -184,7 +184,17 @@ class HTMLParser(_markupbase.ParserBase):
                 elif startswith("<?", i):
                     k = self.parse_pi(i)
                 elif startswith("<!", i):
-                    k = self.parse_declaration(i)
+                    # this might fail with things like <! not a comment > or
+                    # <! -- space before '--' -->.  When strict is True an
+                    # error is raised, when it's False they will be considered
+                    # as bogus comments and parsed (see parse_bogus_comment).
+                    if self.strict:
+                        k = self.parse_declaration(i)
+                    else:
+                        try:
+                            k = self.parse_declaration(i)
+                        except HTMLParseError:
+                            k = self.parse_bogus_comment(i)
                 elif (i + 1) < n:
                     self.handle_data("<")
                     k = i + 1
@@ -256,6 +266,19 @@ class HTMLParser(_markupbase.ParserBase):
             i = self.updatepos(i, n)
         self.rawdata = rawdata[i:]
 
+    # Internal -- parse bogus comment, return length or -1 if not terminated
+    # see http://www.w3.org/TR/html5/tokenization.html#bogus-comment-state
+    def parse_bogus_comment(self, i, report=1):
+        rawdata = self.rawdata
+        if rawdata[i:i+2] != '<!':
+            self.error('unexpected call to parse_comment()')
+        pos = rawdata.find('>', i+2)
+        if pos == -1:
+            return -1
+        if report:
+            self.handle_comment(rawdata[i+2:pos])
+        return pos + 1
+
     # Internal -- parse processing instr, return end or -1 if not terminated
     def parse_pi(self, i):
         rawdata = self.rawdata
diff --git a/Lib/test/test_htmlparser.py b/Lib/test/test_htmlparser.py
index 8c2e25e61a6..7af91311086 100644
--- a/Lib/test/test_htmlparser.py
+++ b/Lib/test/test_htmlparser.py
@@ -323,6 +323,23 @@ DOCTYPE html [
                                 ("endtag", element_lower)],
                             collector=Collector())
 
+    def test_comments(self):
+        html = ("<!-- I'm a valid comment -->"
+                '<!--me too!-->'
+                '<!------>'
+                '<!---->'
+                '<!----I have many hyphens---->'
+                '<!-- I have a > in the middle -->'
+                '<!-- and I have -- in the middle! -->')
+        expected = [('comment', " I'm a valid comment "),
+                    ('comment', 'me too!'),
+                    ('comment', '--'),
+                    ('comment', ''),
+                    ('comment', '--I have many hyphens--'),
+                    ('comment', ' I have a > in the middle '),
+                    ('comment', ' and I have -- in the middle! ')]
+        self._run_check(html, expected)
+
     def test_condcoms(self):
         html = ('<!--[if IE & !(lte IE 8)]>aren\'t<![endif]-->'
                 '<!--[if IE 8]>condcoms<![endif]-->'
@@ -426,6 +443,19 @@ class HTMLParserTolerantTestCase(HTMLParserStrictTestCase):
         # see #12888
         self.assertEqual(p.unescape('&#123; ' * 1050), '{ ' * 1050)
 
+    def test_broken_comments(self):
+        html = ('<! not really a comment >'
+                '<! not a comment either -->'
+                '<! -- close enough -->'
+                '<!!! another bogus comment !!!>')
+        expected = [
+            ('comment', ' not really a comment '),
+            ('comment', ' not a comment either --'),
+            ('comment', ' -- close enough --'),
+            ('comment', '!! another bogus comment !!!'),
+        ]
+        self._run_check(html, expected)
+
     def test_broken_condcoms(self):
         # these condcoms are missing the '--' after '<!' and before the '>'
         html = ('<![if !(IE)]>broken condcom<![endif]>'
diff --git a/Misc/NEWS b/Misc/NEWS
index a14fe6a989c..e2fa04ff951 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -466,6 +466,9 @@ Core and Builtins
 Library
 -------
 
+- Issue #13960: HTMLParser is now able to handle broken comments when
+  strict=False.
+
 - Issue #13921: Undocument and clean up sqlite3.OptimizedUnicode,
   which is obsolete in Python 3.x. It's now aliased to str for
   backwards compatibility.
@@ -498,7 +501,7 @@ Library
 
 - Issue #10881: Fix test_site failure with OS X framework builds.
 
-- Issue #964437 Make IDLE help window non-modal.
+- Issue #964437: Make IDLE help window non-modal.
   Patch by Guilherme Polo and Roger Serwy.
 
 - Issue #13734: Add os.fwalk(), a directory walking function yielding file