diff --git a/azure-pipelines.yml b/azure-pipelines.yml
index 113ad3e338952..b1091ea7f60e4 100644
--- a/azure-pipelines.yml
+++ b/azure-pipelines.yml
@@ -26,3 +26,28 @@ jobs:
   parameters:
     name: Windows
     vmImage: vs2017-win2016
+
+- job: py37_32bit
+  pool:
+    vmImage: ubuntu-18.04
+
+  steps:
+    - script: |
+        docker pull quay.io/pypa/manylinux2014_i686
+        docker run -v $(pwd):/pandas quay.io/pypa/manylinux2014_i686 \
+        /bin/bash -xc "cd pandas && \
+        /opt/python/cp37-cp37m/bin/python -m venv ~/virtualenvs/pandas-dev && \
+        . ~/virtualenvs/pandas-dev/bin/activate && \
+        python -m pip install --no-deps -U pip wheel setuptools && \
+        pip install cython numpy python-dateutil pytz pytest pytest-xdist hypothesis pytest-azurepipelines && \
+        python setup.py build_ext -q -i -j2 && \
+        python -m pip install --no-build-isolation -e . && \
+        pytest -m 'not slow and not network and not clipboard' pandas --junitxml=test-data.xml"
+      displayName: 'Run 32-bit manylinux2014 Docker Build / Tests'
+
+    - task: PublishTestResults@2
+      condition: succeededOrFailed()
+      inputs:
+        testResultsFiles: '**/test-*.xml'
+        failTaskOnFailedTests: true
+        testRunTitle: 'Publish test results for Python 3.7-32 bit full Linux'
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index ebce5b0ef0a66..bdb283ae445b1 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -1255,8 +1255,8 @@ def test_groupby_nat_exclude():
     assert grouped.ngroups == 2
 
     expected = {
-        Timestamp("2013-01-01 00:00:00"): np.array([1, 7], dtype=np.int64),
-        Timestamp("2013-02-01 00:00:00"): np.array([3, 5], dtype=np.int64),
+        Timestamp("2013-01-01 00:00:00"): np.array([1, 7], dtype=np.intp),
+        Timestamp("2013-02-01 00:00:00"): np.array([3, 5], dtype=np.intp),
     }
 
     for k in grouped.indices:
diff --git a/pandas/tests/io/formats/test_info.py b/pandas/tests/io/formats/test_info.py
index 5ef2ce9c47236..6d2d9be97d7fa 100644
--- a/pandas/tests/io/formats/test_info.py
+++ b/pandas/tests/io/formats/test_info.py
@@ -7,7 +7,7 @@
 import numpy as np
 import pytest
 
-from pandas.compat import PYPY
+from pandas.compat import IS64, PYPY
 
 from pandas import (
     CategoricalIndex,
@@ -405,6 +405,7 @@ def test_info_categorical():
     df.info(buf=buf)
 
 
+@pytest.mark.xfail(not IS64, reason="GH 36579: fail on 32-bit system")
 def test_info_int_columns():
     # GH#37245
     df = DataFrame({1: [1, 2], 2: [2, 3]}, index=["A", "B"])