diff --git a/include/pybind11/eigen.h b/include/pybind11/eigen.h
index c0363827c3..d2aac58db9 100644
--- a/include/pybind11/eigen.h
+++ b/include/pybind11/eigen.h
@@ -17,9 +17,23 @@
 
 #include "numpy.h"
 
+// The C4127 suppression was introduced for Eigen 3.4.0. In theory we could
+// make it version specific, or even remove it later, but considering that
+// 1. C4127 is generally far more distracting than useful for modern template code, and
+// 2. we definitely want to ignore any MSVC warnings originating from Eigen code,
+// it is probably best to keep this around indefinitely.
+#if defined(_MSC_VER)
+#  pragma warning(push)
+#  pragma warning(disable: 4127)
+#endif
+
 #include <Eigen/Core>
 #include <Eigen/SparseCore>
 
+#if defined(_MSC_VER)
+#  pragma warning(pop)
+#endif
+
 // Eigen prior to 3.2.7 doesn't have proper move constructors--but worse, some classes get implicit
 // move constructors that break things.  We could detect this an explicitly copy, but an extra copy
 // of matrices seems highly undesirable.
diff --git a/tests/CMakeLists.txt b/tests/CMakeLists.txt
index 6ebcba9b6b..d4e7b71e1d 100644
--- a/tests/CMakeLists.txt
+++ b/tests/CMakeLists.txt
@@ -174,9 +174,9 @@ set(PYBIND11_CROSS_MODULE_GIL_TESTS test_gil_scoped.py)
 set(PYBIND11_EIGEN_REPO
     "https://gitlab.com/libeigen/eigen.git"
     CACHE STRING "Eigen repository to use for tests")
-# This hash is for 3.3.8, using a hash for security reasons
+# This hash is for 3.4.0, using a hash for security reasons
 set(PYBIND11_EIGEN_VERSION
-    "dc252fbf00079ccab57948a164b1421703fe4361"
+    "929bc0e191d0927b1735b9a1ddc0e8b77e3a25ec"
     CACHE STRING "Eigen version to use for tests")
 
 # Check if Eigen is available; if not, remove from PYBIND11_TEST_FILES (but