diff --git a/Doc/library/pathlib.rst b/Doc/library/pathlib.rst
index dead49b630dcdf..a68faca98c903f 100644
--- a/Doc/library/pathlib.rst
+++ b/Doc/library/pathlib.rst
@@ -536,7 +536,7 @@ Pure paths provide the following methods and properties:
       True
 
 
-.. method:: PurePath.relative_to(*other)
+.. method:: PurePath.relative_to(*other, strict=True)
 
    Compute a version of this path relative to the path represented by
    *other*.  If it's impossible, ValueError is raised::
@@ -549,9 +549,25 @@ Pure paths provide the following methods and properties:
       >>> p.relative_to('/usr')
       Traceback (most recent call last):
         File "<stdin>", line 1, in <module>
-        File "pathlib.py", line 694, in relative_to
-          .format(str(self), str(formatted)))
+        File "pathlib.py", line 940, in relative_to
+          raise ValueError(error_message.format(str(self), str(formatted)))
       ValueError: '/etc/passwd' does not start with '/usr'
+      >>> p.relative_to('/usr', strict=False)
+      PurePosixPath('../etc/passwd')
+      >>> p.relative_to('foo', strict=False)
+      Traceback (most recent call last):
+        File "<stdin>", line 1, in <module>
+        File "pathlib.py", line 940, in relative_to
+          raise ValueError(error_message.format(str(self), str(formatted)))
+      ValueError: '/etc/passwd' is not related to 'foo'
+
+   If the path doesn't start with *other* and *strict* is ``True``,
+   :exc:`ValueError` is raised.  If *strict* is ``False`` and the paths are
+   not both relative or both absolute :exc:`ValueError` is raised (on Windows
+   both paths must reference the same drive as well).
+
+   .. versionadded:: 3.9
+      The *strict* parameter was added.
 
 
 .. method:: PurePath.with_name(name)
diff --git a/Lib/pathlib.py b/Lib/pathlib.py
index f98d69eb04ac31..67e4c400937b5f 100644
--- a/Lib/pathlib.py
+++ b/Lib/pathlib.py
@@ -895,10 +895,10 @@ def with_suffix(self, suffix):
         return self._from_parsed_parts(self._drv, self._root,
                                        self._parts[:-1] + [name])
 
-    def relative_to(self, *other):
+    def relative_to(self, *other, strict=True):
         """Return the relative path to another path identified by the passed
         arguments.  If the operation is not possible (because this is not
-        a subpath of the other path), raise ValueError.
+        related to the other path), raise ValueError.
         """
         # For the purpose of this method, drive and root are considered
         # separate parts, i.e.:
@@ -920,12 +920,26 @@ def relative_to(self, *other):
             to_abs_parts = to_parts
         n = len(to_abs_parts)
         cf = self._flavour.casefold_parts
-        if (root or drv) if n == 0 else cf(abs_parts[:n]) != cf(to_abs_parts):
+        common = 0
+        for p, tp in zip(cf(abs_parts), cf(to_abs_parts)):
+            if p != tp:
+                break
+            common += 1
+        if strict:
+            failure = (root or drv) if n == 0 else common != n
+            error_message = "{!r} does not start with {!r}"
+            up_parts = []
+        else:
+            failure = root != to_root
+            if drv or to_drv:
+                failure = cf([drv]) != cf([to_drv]) or (failure and n > 1)
+            error_message = "{!r} is not related to {!r}"
+            up_parts = (n-common)*['..']
+        if failure:
             formatted = self._format_parsed_parts(to_drv, to_root, to_parts)
-            raise ValueError("{!r} does not start with {!r}"
-                             .format(str(self), str(formatted)))
-        return self._from_parsed_parts('', root if n == 1 else '',
-                                       abs_parts[n:])
+            raise ValueError(error_message.format(str(self), str(formatted)))
+        return self._from_parsed_parts('', root if common == 1 else '',
+                                       up_parts+abs_parts[common:])
 
     def is_relative_to(self, *other):
         """Return True if the path is relative to another path or False.
diff --git a/Lib/test/test_pathlib.py b/Lib/test/test_pathlib.py
index 1589282886b6b8..5324f9b0e2ca10 100644
--- a/Lib/test/test_pathlib.py
+++ b/Lib/test/test_pathlib.py
@@ -613,13 +613,29 @@ def test_relative_to_common(self):
         self.assertEqual(p.relative_to('a/'), P('b'))
         self.assertEqual(p.relative_to(P('a/b')), P())
         self.assertEqual(p.relative_to('a/b'), P())
+        self.assertEqual(p.relative_to(P(), strict=False), P('a/b'))
+        self.assertEqual(p.relative_to('', strict=False), P('a/b'))
+        self.assertEqual(p.relative_to(P('a'), strict=False), P('b'))
+        self.assertEqual(p.relative_to('a', strict=False), P('b'))
+        self.assertEqual(p.relative_to('a/', strict=False), P('b'))
+        self.assertEqual(p.relative_to(P('a/b'), strict=False), P())
+        self.assertEqual(p.relative_to('a/b', strict=False), P())
+        self.assertEqual(p.relative_to(P('a/c'), strict=False), P('../b'))
+        self.assertEqual(p.relative_to('a/c', strict=False), P('../b'))
+        self.assertEqual(p.relative_to(P('a/b/c'), strict=False), P('..'))
+        self.assertEqual(p.relative_to('a/b/c', strict=False), P('..'))
+        self.assertEqual(p.relative_to(P('c'), strict=False), P('../a/b'))
+        self.assertEqual(p.relative_to('c', strict=False), P('../a/b'))
         # With several args.
         self.assertEqual(p.relative_to('a', 'b'), P())
+        self.assertEqual(p.relative_to('a', 'b', strict=False), P())
         # Unrelated paths.
         self.assertRaises(ValueError, p.relative_to, P('c'))
         self.assertRaises(ValueError, p.relative_to, P('a/b/c'))
         self.assertRaises(ValueError, p.relative_to, P('a/c'))
         self.assertRaises(ValueError, p.relative_to, P('/a'))
+        self.assertRaises(ValueError, p.relative_to, P('/'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('/a'), strict=False)
         p = P('/a/b')
         self.assertEqual(p.relative_to(P('/')), P('a/b'))
         self.assertEqual(p.relative_to('/'), P('a/b'))
@@ -628,6 +644,19 @@ def test_relative_to_common(self):
         self.assertEqual(p.relative_to('/a/'), P('b'))
         self.assertEqual(p.relative_to(P('/a/b')), P())
         self.assertEqual(p.relative_to('/a/b'), P())
+        self.assertEqual(p.relative_to(P('/'), strict=False), P('a/b'))
+        self.assertEqual(p.relative_to('/', strict=False), P('a/b'))
+        self.assertEqual(p.relative_to(P('/a'), strict=False), P('b'))
+        self.assertEqual(p.relative_to('/a', strict=False), P('b'))
+        self.assertEqual(p.relative_to('/a/', strict=False), P('b'))
+        self.assertEqual(p.relative_to(P('/a/b'), strict=False), P())
+        self.assertEqual(p.relative_to('/a/b', strict=False), P())
+        self.assertEqual(p.relative_to(P('/a/c'), strict=False), P('../b'))
+        self.assertEqual(p.relative_to('/a/c', strict=False), P('../b'))
+        self.assertEqual(p.relative_to(P('/a/b/c'), strict=False), P('..'))
+        self.assertEqual(p.relative_to('/a/b/c', strict=False), P('..'))
+        self.assertEqual(p.relative_to(P('/c'), strict=False), P('../a/b'))
+        self.assertEqual(p.relative_to('/c', strict=False), P('../a/b'))
         # Unrelated paths.
         self.assertRaises(ValueError, p.relative_to, P('/c'))
         self.assertRaises(ValueError, p.relative_to, P('/a/b/c'))
@@ -635,6 +664,8 @@ def test_relative_to_common(self):
         self.assertRaises(ValueError, p.relative_to, P())
         self.assertRaises(ValueError, p.relative_to, '')
         self.assertRaises(ValueError, p.relative_to, P('a'))
+        self.assertRaises(ValueError, p.relative_to, P(''), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('a'), strict=False)
 
     def test_is_relative_to_common(self):
         P = self.cls
@@ -1079,6 +1110,16 @@ def test_relative_to(self):
         self.assertEqual(p.relative_to('c:foO/'), P('Bar'))
         self.assertEqual(p.relative_to(P('c:foO/baR')), P())
         self.assertEqual(p.relative_to('c:foO/baR'), P())
+        self.assertEqual(p.relative_to(P('c:'), strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to('c:', strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to(P('c:foO'), strict=False), P('Bar'))
+        self.assertEqual(p.relative_to('c:foO', strict=False), P('Bar'))
+        self.assertEqual(p.relative_to('c:foO/', strict=False), P('Bar'))
+        self.assertEqual(p.relative_to(P('c:foO/baR'), strict=False), P())
+        self.assertEqual(p.relative_to('c:foO/baR', strict=False), P())
+        self.assertEqual(p.relative_to(P('C:Foo/Bar/Baz'), strict=False), P('..'))
+        self.assertEqual(p.relative_to(P('C:Foo/Baz'), strict=False), P('../Bar'))
+        self.assertEqual(p.relative_to(P('C:Baz/Bar'), strict=False), P('../../Foo/Bar'))
         # Unrelated paths.
         self.assertRaises(ValueError, p.relative_to, P())
         self.assertRaises(ValueError, p.relative_to, '')
@@ -1089,6 +1130,13 @@ def test_relative_to(self):
         self.assertRaises(ValueError, p.relative_to, P('C:/Foo'))
         self.assertRaises(ValueError, p.relative_to, P('C:Foo/Bar/Baz'))
         self.assertRaises(ValueError, p.relative_to, P('C:Foo/Baz'))
+        self.assertRaises(ValueError, p.relative_to, P(), strict=False)
+        self.assertRaises(ValueError, p.relative_to, '', strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('d:'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('/'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('/Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('C:/Foo'), strict=False)
         p = P('C:/Foo/Bar')
         self.assertEqual(p.relative_to(P('c:')), P('/Foo/Bar'))
         self.assertEqual(p.relative_to('c:'), P('/Foo/Bar'))
@@ -1101,6 +1149,20 @@ def test_relative_to(self):
         self.assertEqual(p.relative_to('c:/foO/'), P('Bar'))
         self.assertEqual(p.relative_to(P('c:/foO/baR')), P())
         self.assertEqual(p.relative_to('c:/foO/baR'), P())
+        self.assertEqual(p.relative_to(P('c:'), strict=False), P('/Foo/Bar'))
+        self.assertEqual(p.relative_to('c:', strict=False), P('/Foo/Bar'))
+        self.assertEqual(str(p.relative_to(P('c:'), strict=False)), '\\Foo\\Bar')
+        self.assertEqual(str(p.relative_to('c:', strict=False)), '\\Foo\\Bar')
+        self.assertEqual(p.relative_to(P('c:/'), strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to('c:/', strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to(P('c:/foO'), strict=False), P('Bar'))
+        self.assertEqual(p.relative_to('c:/foO', strict=False), P('Bar'))
+        self.assertEqual(p.relative_to('c:/foO/', strict=False), P('Bar'))
+        self.assertEqual(p.relative_to(P('c:/foO/baR'), strict=False), P())
+        self.assertEqual(p.relative_to('c:/foO/baR', strict=False), P())
+        self.assertEqual(p.relative_to('C:/Baz', strict=False), P('../Foo/Bar'))
+        self.assertEqual(p.relative_to('C:/Foo/Bar/Baz', strict=False), P('..'))
+        self.assertEqual(p.relative_to('C:/Foo/Baz', strict=False), P('../Bar'))
         # Unrelated paths.
         self.assertRaises(ValueError, p.relative_to, P('C:/Baz'))
         self.assertRaises(ValueError, p.relative_to, P('C:/Foo/Bar/Baz'))
@@ -1111,6 +1173,12 @@ def test_relative_to(self):
         self.assertRaises(ValueError, p.relative_to, P('/'))
         self.assertRaises(ValueError, p.relative_to, P('/Foo'))
         self.assertRaises(ValueError, p.relative_to, P('//C/Foo'))
+        self.assertRaises(ValueError, p.relative_to, P('C:Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('d:'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('d:/'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('/'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('/Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('//C/Foo'), strict=False)
         # UNC paths.
         p = P('//Server/Share/Foo/Bar')
         self.assertEqual(p.relative_to(P('//sErver/sHare')), P('Foo/Bar'))
@@ -1121,11 +1189,25 @@ def test_relative_to(self):
         self.assertEqual(p.relative_to('//sErver/sHare/Foo/'), P('Bar'))
         self.assertEqual(p.relative_to(P('//sErver/sHare/Foo/Bar')), P())
         self.assertEqual(p.relative_to('//sErver/sHare/Foo/Bar'), P())
+        self.assertEqual(p.relative_to(P('//sErver/sHare'), strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to('//sErver/sHare', strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to('//sErver/sHare/', strict=False), P('Foo/Bar'))
+        self.assertEqual(p.relative_to(P('//sErver/sHare/Foo'), strict=False), P('Bar'))
+        self.assertEqual(p.relative_to('//sErver/sHare/Foo', strict=False), P('Bar'))
+        self.assertEqual(p.relative_to('//sErver/sHare/Foo/', strict=False), P('Bar'))
+        self.assertEqual(p.relative_to(P('//sErver/sHare/Foo/Bar'), strict=False), P())
+        self.assertEqual(p.relative_to('//sErver/sHare/Foo/Bar', strict=False), P())
+        self.assertEqual(p.relative_to(P('//sErver/sHare/bar'), strict=False), P('../Foo/Bar'))
+        self.assertEqual(p.relative_to('//sErver/sHare/bar', strict=False), P('../Foo/Bar'))
         # Unrelated paths.
         self.assertRaises(ValueError, p.relative_to, P('/Server/Share/Foo'))
         self.assertRaises(ValueError, p.relative_to, P('c:/Server/Share/Foo'))
         self.assertRaises(ValueError, p.relative_to, P('//z/Share/Foo'))
         self.assertRaises(ValueError, p.relative_to, P('//Server/z/Foo'))
+        self.assertRaises(ValueError, p.relative_to, P('/Server/Share/Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('c:/Server/Share/Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('//z/Share/Foo'), strict=False)
+        self.assertRaises(ValueError, p.relative_to, P('//Server/z/Foo'), strict=False)
 
     def test_is_relative_to(self):
         P = self.cls
diff --git a/Misc/NEWS.d/next/Library/2020-04-30-02-15-08.bpo-40358.A4ygqe.rst b/Misc/NEWS.d/next/Library/2020-04-30-02-15-08.bpo-40358.A4ygqe.rst
new file mode 100644
index 00000000000000..219c41ae4e5f07
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2020-04-30-02-15-08.bpo-40358.A4ygqe.rst
@@ -0,0 +1,2 @@
+Add strict parameter in :meth:`pathlib.PurePath.relative_to`. Patch by
+Domenico Ragusa.