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/arrays/floating/test_function.py b/pandas/tests/arrays/floating/test_function.py
index 2767d93741d4c..baf60a363ad29 100644
--- a/pandas/tests/arrays/floating/test_function.py
+++ b/pandas/tests/arrays/floating/test_function.py
@@ -1,6 +1,8 @@
 import numpy as np
 import pytest
 
+from pandas.compat import IS64
+
 import pandas as pd
 import pandas._testing as tm
 
@@ -71,6 +73,7 @@ def test_ufunc_reduce_raises(values):
         np.add.reduce(a)
 
 
+@pytest.mark.skipif(not IS64, reason="GH 36579: fail on 32-bit system")
 @pytest.mark.parametrize(
     "pandasmethname, kwargs",
     [
diff --git a/pandas/tests/base/test_misc.py b/pandas/tests/base/test_misc.py
index f7952c81cfd61..6a9d58021a4d9 100644
--- a/pandas/tests/base/test_misc.py
+++ b/pandas/tests/base/test_misc.py
@@ -3,7 +3,7 @@
 import numpy as np
 import pytest
 
-from pandas.compat import PYPY
+from pandas.compat import IS64, PYPY
 
 from pandas.core.dtypes.common import (
     is_categorical_dtype,
@@ -128,7 +128,10 @@ def test_memory_usage(index_or_series_obj):
     )
 
     if len(obj) == 0:
-        expected = 0 if isinstance(obj, Index) else 80
+        if isinstance(obj, Index):
+            expected = 0
+        else:
+            expected = 80 if IS64 else 48
         assert res_deep == res == expected
     elif is_object or is_categorical:
         # only deep will pick them up
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index 2e51fca71e139..b57fa2540add9 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -1268,8 +1268,8 @@ 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 418d05a6b8752..8c2155aec7248 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,
@@ -475,6 +475,7 @@ 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"])
diff --git a/pandas/tests/io/parser/test_c_parser_only.py b/pandas/tests/io/parser/test_c_parser_only.py
index ae63b6af3a8b6..eee111dd4579c 100644
--- a/pandas/tests/io/parser/test_c_parser_only.py
+++ b/pandas/tests/io/parser/test_c_parser_only.py
@@ -13,6 +13,7 @@
 import numpy as np
 import pytest
 
+from pandas.compat import IS64
 from pandas.errors import ParserError
 import pandas.util._test_decorators as td
 
@@ -717,7 +718,10 @@ def test_float_precision_options(c_parser_only):
 
     df3 = parser.read_csv(StringIO(s), float_precision="legacy")
 
-    assert not df.iloc[0, 0] == df3.iloc[0, 0]
+    if IS64:
+        assert not df.iloc[0, 0] == df3.iloc[0, 0]
+    else:
+        assert df.iloc[0, 0] == df3.iloc[0, 0]
 
     msg = "Unrecognized float_precision option: junk"
 
diff --git a/pandas/tests/reshape/test_pivot.py b/pandas/tests/reshape/test_pivot.py
index 92128def4540a..642e6a691463e 100644
--- a/pandas/tests/reshape/test_pivot.py
+++ b/pandas/tests/reshape/test_pivot.py
@@ -4,6 +4,8 @@
 import numpy as np
 import pytest
 
+from pandas.compat import IS64
+
 import pandas as pd
 from pandas import (
     Categorical,
@@ -2104,6 +2106,7 @@ def test_pivot_duplicates(self):
         with pytest.raises(ValueError, match="duplicate entries"):
             data.pivot("a", "b", "c")
 
+    @pytest.mark.xfail(not IS64, reason="GH 36579: fail on 32-bit system")
     def test_pivot_empty(self):
         df = DataFrame(columns=["a", "b", "c"])
         result = df.pivot("a", "b", "c")