diff --git a/Lib/test/test_time.py b/Lib/test/test_time.py
index 4e31abf4ec8eec7..1008380a287ebfe 100644
--- a/Lib/test/test_time.py
+++ b/Lib/test/test_time.py
@@ -120,7 +120,13 @@ def test_clock_monotonic(self):
     def test_pthread_getcpuclockid(self):
         clk_id = time.pthread_getcpuclockid(threading.get_ident())
         self.assertTrue(type(clk_id) is int)
-        self.assertNotEqual(clk_id, time.CLOCK_THREAD_CPUTIME_ID)
+        # when in 32-bit mode AIX only returns the predefined constant
+        if not platform.system() == "AIX":
+            self.assertNotEqual(clk_id, time.CLOCK_THREAD_CPUTIME_ID)
+        elif (sys.maxsize.bit_length() > 32):
+            self.assertNotEqual(clk_id, time.CLOCK_THREAD_CPUTIME_ID)
+        else:
+            self.assertEqual(clk_id, time.CLOCK_THREAD_CPUTIME_ID)
         t1 = time.clock_gettime(clk_id)
         t2 = time.clock_gettime(clk_id)
         self.assertLessEqual(t1, t2)
diff --git a/Misc/NEWS.d/next/Library/2019-04-08-14-41-22.bpo-34373.lEAl_-.rst b/Misc/NEWS.d/next/Library/2019-04-08-14-41-22.bpo-34373.lEAl_-.rst
new file mode 100644
index 000000000000000..af593d4925b7298
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2019-04-08-14-41-22.bpo-34373.lEAl_-.rst
@@ -0,0 +1,2 @@
+Fix :func:`time.mktime` error handling on AIX for year before 1970.
+Fix `test_time.test_pthread_getcpuclockid` for AIX in 32-bit mode.
diff --git a/Modules/timemodule.c b/Modules/timemodule.c
index 4c8e2cb2344bcd2..3cc40767ab440a4 100644
--- a/Modules/timemodule.c
+++ b/Modules/timemodule.c
@@ -960,44 +960,68 @@ not present, current time as returned by localtime() is used.");
 
 #ifdef HAVE_MKTIME
 static PyObject *
-time_mktime(PyObject *self, PyObject *tup)
+time_mktime(PyObject *self, PyObject *tm_tuple)
 {
-    struct tm buf;
+    struct tm tm;
     time_t tt;
-    if (!gettmarg(tup, &buf,
+
+    if (!gettmarg(tm_tuple, &tm,
                   "iiiiiiiii;mktime(): illegal time tuple argument"))
     {
         return NULL;
     }
+
 #ifdef _AIX
-    /* year < 1902 or year > 2037 */
-    if (buf.tm_year < 2 || buf.tm_year > 137) {
+    /* bpo-19748: AIX mktime() valid range is 00:00:00 UTC, January 1, 1970
+       to 03:14:07 UTC, January 19, 2038. Thanks to the workaround below,
+       it is possible to support years in range [1902; 2037] */
+    if (tm.tm_year < 2 || tm.tm_year > 137) {
         /* Issue #19748: On AIX, mktime() doesn't report overflow error for
          * timestamp < -2^31 or timestamp > 2**31-1. */
         PyErr_SetString(PyExc_OverflowError,
                         "mktime argument out of range");
         return NULL;
     }
-#else
-    buf.tm_wday = -1;  /* sentinel; original value ignored */
+
+    /* bpo-34373: AIX mktime() has an integer overflow for years in range
+       [1902; 1969]. Workaround the issue by using a year greater or equal than
+       1970 (tm_year >= 70): mktime() behaves correctly in that case
+       (ex: properly report errors). tm_year and tm_wday are adjusted after
+       mktime() call. */
+    int orig_tm_year = tm.tm_year;
+    int delta_days = 0;
+    while (tm.tm_year < 70) {
+        /* Use 4 years to account properly leap years */
+        tm.tm_year += 4;
+        delta_days -= (366 + (365 * 3));
+    }
 #endif
-    tt = mktime(&buf);
+
+    tm.tm_wday = -1;  /* sentinel; original value ignored */
+    tt = mktime(&tm);
+
     /* Return value of -1 does not necessarily mean an error, but tm_wday
      * cannot remain set to -1 if mktime succeeded. */
     if (tt == (time_t)(-1)
-#ifndef _AIX
         /* Return value of -1 does not necessarily mean an error, but
          * tm_wday cannot remain set to -1 if mktime succeeded. */
-        && buf.tm_wday == -1
-#else
-        /* on AIX, tm_wday is always sets, even on error */
-#endif
-       )
+        && tm.tm_wday == -1)
     {
         PyErr_SetString(PyExc_OverflowError,
                         "mktime argument out of range");
         return NULL;
     }
+
+#ifdef _AIX
+    if (delta_days != 0) {
+        tm.tm_year = orig_tm_year;
+        if (tm.tm_wday != -1) {
+            tm.tm_wday = (tm.tm_wday + delta_days) % 7;
+        }
+        tt += delta_days * (24 * 3600);
+    }
+#endif
+
     return PyFloat_FromDouble((double)tt);
 }
 
diff --git a/Python/pytime.c b/Python/pytime.c
index 0e9413174195d9b..9ff300699f04af0 100644
--- a/Python/pytime.c
+++ b/Python/pytime.c
@@ -1062,12 +1062,23 @@ _PyTime_localtime(time_t t, struct tm *tm)
     }
     return 0;
 #else /* !MS_WINDOWS */
+
+#ifdef _AIX
+    /* bpo-34373: AIX does not return NULL if t is too small or too large */
+    if (t < -2145916800 /* 1902-01-01 */
+       || t > 2145916800 /* 2038-01-01 */) {
+        errno = EINVAL;
+        PyErr_SetString(PyExc_OverflowError,
+                        "localtime argument out of range");
+        return -1;
+    }
+#endif
+
+    errno = 0;
     if (localtime_r(&t, tm) == NULL) {
-#ifdef EINVAL
         if (errno == 0) {
             errno = EINVAL;
         }
-#endif
         PyErr_SetFromErrno(PyExc_OSError);
         return -1;
     }