diff --git a/appveyor.yml b/appveyor.yml
index 082063ae572..3c4c2c3dc76 100644
--- a/appveyor.yml
+++ b/appveyor.yml
@@ -36,14 +36,6 @@ cache:
 
 matrix:
   fast_finish: true
-  allow_failures:
-    # Can't build with 32-bit MinGW for now.
-    # See https://issues.apache.org/jira/browse/ARROW-4297
-    - JOB: "MinGW32"
-      MINGW_PACKAGE_PREFIX: mingw-w64-i686
-      MINGW_PREFIX: c:\msys64\mingw32
-      MSYSTEM: MINGW32
-      USE_CLCACHE: false
 
 environment:
   global:
diff --git a/ci/appveyor-cpp-build-mingw.bat b/ci/appveyor-cpp-build-mingw.bat
index 4d399274549..bdd3b14462e 100644
--- a/ci/appveyor-cpp-build-mingw.bat
+++ b/ci/appveyor-cpp-build-mingw.bat
@@ -44,8 +44,11 @@ cmake ^
     -DARROW_PYTHON=ON ^
     -DPythonInterp_FIND_VERSION=ON ^
     -DPythonInterp_FIND_VERSION_MAJOR=3 ^
+    -DARROW_BUILD_TESTS=ON ^
+    -DARROW_PYTHON=OFF ^
     .. || exit /B
 make -j4 || exit /B
+ctest --output-on-failure -j2 || exit /B
 make install || exit /B
 popd
 
diff --git a/ci/appveyor-cpp-setup-mingw.bat b/ci/appveyor-cpp-setup-mingw.bat
index 471e7426f6e..0c3d633cf8a 100644
--- a/ci/appveyor-cpp-setup-mingw.bat
+++ b/ci/appveyor-cpp-setup-mingw.bat
@@ -25,6 +25,7 @@ pacman -S --noconfirm ^
     "%MINGW_PACKAGE_PREFIX%-cmake" ^
     "%MINGW_PACKAGE_PREFIX%-flatbuffers" ^
     "%MINGW_PACKAGE_PREFIX%-gcc" ^
+    "%MINGW_PACKAGE_PREFIX%-gflags" ^
     "%MINGW_PACKAGE_PREFIX%-gobject-introspection" ^
     "%MINGW_PACKAGE_PREFIX%-gtk-doc" ^
     "%MINGW_PACKAGE_PREFIX%-lz4" ^
diff --git a/cpp/src/arrow/CMakeLists.txt b/cpp/src/arrow/CMakeLists.txt
index 94eba0ce1a6..9ddadc5a555 100644
--- a/cpp/src/arrow/CMakeLists.txt
+++ b/cpp/src/arrow/CMakeLists.txt
@@ -301,13 +301,24 @@ arrow_add_pkg_config("arrow")
 #
 
 add_arrow_test(allocator-test)
-add_arrow_test(array-test
-               SOURCES
-               array-test.cc
-               array-binary-test.cc
-               array-dict-test.cc
-               array-list-test.cc
-               array-struct-test.cc)
+
+if(WIN32)
+  add_arrow_test(array-test
+                 SOURCES
+                 array-test.cc
+                 array-binary-test.cc
+                 array-list-test.cc
+                 array-struct-test.cc)
+else()
+  add_arrow_test(array-test
+                 SOURCES
+                 array-test.cc
+                 array-binary-test.cc
+                 array-dict-test.cc
+                 array-list-test.cc
+                 array-struct-test.cc)
+endif()
+
 add_arrow_test(buffer-test)
 
 if(ARROW_IPC)
diff --git a/cpp/src/arrow/compute/kernels/cast-test.cc b/cpp/src/arrow/compute/kernels/cast-test.cc
index e7f5a4aa296..0fd3065bd9b 100644
--- a/cpp/src/arrow/compute/kernels/cast-test.cc
+++ b/cpp/src/arrow/compute/kernels/cast-test.cc
@@ -424,6 +424,7 @@ TEST_F(TestCast, FloatingPointToInt) {
                                                     options);
 }
 
+#if ARROW_BITNESS >= 64
 TEST_F(TestCast, IntToFloatingPoint) {
   auto options = CastOptions::Safe();
 
@@ -438,6 +439,7 @@ TEST_F(TestCast, IntToFloatingPoint) {
                                                     UnsafeVectorCast<double, int64_t>(v1),
                                                     options);
 }
+#endif
 
 TEST_F(TestCast, TimestampToTimestamp) {
   CastOptions options;
diff --git a/cpp/src/arrow/util/hash-util.h b/cpp/src/arrow/util/hash-util.h
index 637f4328085..7aed3c171dc 100644
--- a/cpp/src/arrow/util/hash-util.h
+++ b/cpp/src/arrow/util/hash-util.h
@@ -81,10 +81,13 @@ class HashUtil {
     const uint8_t* p = reinterpret_cast<const uint8_t*>(data);
     const uint8_t* end = p + nbytes;
 
+#if ARROW_BITNESS >= 64
     while (p <= end - 8) {
       hash = HW_crc32_u64(hash, *reinterpret_cast<const uint64_t*>(p));
       p += 8;
     }
+#endif
+
     while (p <= end - 4) {
       hash = HW_crc32_u32(hash, *reinterpret_cast<const uint32_t*>(p));
       p += 4;
@@ -113,6 +116,7 @@ class HashUtil {
     uint32_t h1 = static_cast<uint32_t>(hash >> 32);
     uint32_t h2 = static_cast<uint32_t>(hash);
 
+#if ARROW_BITNESS >= 64
     while (nbytes >= 16) {
       h1 = HW_crc32_u64(h1, *reinterpret_cast<const uint64_t*>(p));
       h2 = HW_crc32_u64(h2, *reinterpret_cast<const uint64_t*>(p + 8));
@@ -125,6 +129,15 @@ class HashUtil {
       nbytes -= 8;
       p += 8;
     }
+#else
+    while (nbytes >= 8) {
+      h1 = HW_crc32_u32(h1, *reinterpret_cast<const uint32_t*>(p));
+      h2 = HW_crc32_u32(h2, *reinterpret_cast<const uint32_t*>(p + 4));
+      nbytes -= 8;
+      p += 8;
+    }
+#endif
+
     if (nbytes >= 4) {
       h1 = HW_crc32_u16(h1, *reinterpret_cast<const uint16_t*>(p));
       h2 = HW_crc32_u16(h2, *reinterpret_cast<const uint16_t*>(p + 2));
@@ -151,58 +164,6 @@ class HashUtil {
     return (static_cast<uint64_t>(h1) << 32) + h2;
   }
 
-  /// CrcHash() specialized for 1-byte data
-  static inline uint32_t CrcHash1(const void* v, uint32_t hash) {
-    const uint8_t* s = reinterpret_cast<const uint8_t*>(v);
-    hash = HW_crc32_u8(hash, *s);
-    hash = (hash << 16) | (hash >> 16);
-    return hash;
-  }
-
-  /// CrcHash() specialized for 2-byte data
-  static inline uint32_t CrcHash2(const void* v, uint32_t hash) {
-    const uint16_t* s = reinterpret_cast<const uint16_t*>(v);
-    hash = HW_crc32_u16(hash, *s);
-    hash = (hash << 16) | (hash >> 16);
-    return hash;
-  }
-
-  /// CrcHash() specialized for 4-byte data
-  static inline uint32_t CrcHash4(const void* v, uint32_t hash) {
-    const uint32_t* p = reinterpret_cast<const uint32_t*>(v);
-    hash = HW_crc32_u32(hash, *p);
-    hash = (hash << 16) | (hash >> 16);
-    return hash;
-  }
-
-  /// CrcHash() specialized for 8-byte data
-  static inline uint32_t CrcHash8(const void* v, uint32_t hash) {
-    const uint64_t* p = reinterpret_cast<const uint64_t*>(v);
-    hash = HW_crc32_u64(hash, *p);
-    hash = (hash << 16) | (hash >> 16);
-    return hash;
-  }
-
-  /// CrcHash() specialized for 12-byte data
-  static inline uint32_t CrcHash12(const void* v, uint32_t hash) {
-    const uint64_t* p = reinterpret_cast<const uint64_t*>(v);
-    hash = HW_crc32_u64(hash, *p);
-    ++p;
-    hash = HW_crc32_u32(hash, *reinterpret_cast<const uint32_t*>(p));
-    hash = (hash << 16) | (hash >> 16);
-    return hash;
-  }
-
-  /// CrcHash() specialized for 16-byte data
-  static inline uint32_t CrcHash16(const void* v, uint32_t hash) {
-    const uint64_t* p = reinterpret_cast<const uint64_t*>(v);
-    hash = HW_crc32_u64(hash, *p);
-    ++p;
-    hash = HW_crc32_u64(hash, *p);
-    hash = (hash << 16) | (hash >> 16);
-    return hash;
-  }
-
   static const uint64_t MURMUR_PRIME = 0xc6a4a7935bd1e995;
   static const int MURMUR_R = 47;
 
diff --git a/cpp/src/arrow/util/macros.h b/cpp/src/arrow/util/macros.h
index 5f1934d732c..4516985e300 100644
--- a/cpp/src/arrow/util/macros.h
+++ b/cpp/src/arrow/util/macros.h
@@ -18,6 +18,8 @@
 #ifndef ARROW_UTIL_MACROS_H
 #define ARROW_UTIL_MACROS_H
 
+#include <cstdint>
+
 #define ARROW_STRINGIFY(x) #x
 #define ARROW_CONCAT(x, y) x##y
 
@@ -123,6 +125,17 @@
 #define ARROW_DISABLE_UBSAN(feature)
 #endif
 
+// ----------------------------------------------------------------------
+// Machine information
+
+#if INTPTR_MAX == INT64_MAX
+#define ARROW_BITNESS 64
+#elif INTPTR_MAX == INT32_MAX
+#define ARROW_BITNESS 32
+#else
+#error Unexpected INTPTR_MAX
+#endif
+
 // ----------------------------------------------------------------------
 // From googletest
 // (also in parquet-cpp)
diff --git a/cpp/src/arrow/util/sse-util.h b/cpp/src/arrow/util/sse-util.h
index edaf6686be5..15e7c99581a 100644
--- a/cpp/src/arrow/util/sse-util.h
+++ b/cpp/src/arrow/util/sse-util.h
@@ -110,7 +110,11 @@ static inline uint32_t SSE4_crc32_u32(uint32_t crc, uint32_t v) {
 }
 
 static inline uint32_t SSE4_crc32_u64(uint32_t crc, uint64_t v) {
+#if ARROW_BITNESS == 32
+  return 0;
+#else
   return static_cast<uint32_t>(_mm_crc32_u64(crc, v));
+#endif
 }
 
 #else  // without SSE 4.2.