diff --git a/.travis.yml b/.travis.yml
index 9169391..a9ee89c 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -4,3 +4,4 @@ node_js:
   - "8" # to be removed on "December 2019"
   - "10" # to be removed on "April 2021"
   - "12" # to be removed on "April 2022"
+  - "13" # to be removed on "June 2020"
diff --git a/README.md b/README.md
index a2b69ba..3a65d00 100644
--- a/README.md
+++ b/README.md
@@ -79,11 +79,11 @@ SPEC
       - when messages does not match, rejects with the actual error.
     - if `error` is a `<Class>` (constructor function), validate instanceof using constructor (works well with ES2015 classes that extends Error).
       - when actual error is an instanceof `<Class>`, resolves with undefined.
-      - when actual error is NOT an instanceof `<Class>`, rejects with the actual error.
+      - when actual error is NOT an instanceof `<Class>`, rejects with AssertionError.
       - appends `error.name` as expected error class name to the message if the `promiseFn` is not rejected.
     - if `error` is a `<Function>`, run custom validation against actual rejection result.
       - when validation function returns `true`, resolves with undefined.
-      - when returned value of validation function is NOT `true`, rejects with the actual error.
+      - when returned value of validation function is NOT `true`, rejects with AssertionError.
       - if Error is thrown from validation function, rejects with the error.
     - if `error` is an `<Object>`, that is an object where each property will be tested for.
       - when all key-value pairs in `error` are the same as key-value pairs from actual rejected result, resolves with undefined. Note that only properties on the error object will be tested.
diff --git a/index.js b/index.js
index 97f7ab6..9e48b7c 100644
--- a/index.js
+++ b/index.js
@@ -96,13 +96,27 @@ function wantReject (stackStartFn, thennable, errorHandler, message) {
             // Dealing with ES2015 class that extends Error
             // see: https://github.com/nodejs/node/issues/3188
             // see: https://github.com/nodejs/node/pull/4166
-            return reject(actualRejectionResult);
+            return reject(new AssertionError({
+              actual: actualRejectionResult,
+              expected: errorHandler,
+              message: message || 'The error is expected to be an instance of "' + errorHandler.name + '". Received "' + actualRejectionResult.constructor.name + '"\n\nError message:\n\n' + actualRejectionResult.message,
+              operator: stackStartFn.name,
+              stackStartFn: stackStartFn
+            }));
           }
         }
-        if (errorHandler.call({}, actualRejectionResult) === true) {
+        var handlerFuncResult = errorHandler.call({}, actualRejectionResult);
+        if (handlerFuncResult === true) {
           return resolve();
         } else {
-          return reject(actualRejectionResult);
+          var validationFunctionName = errorHandler.name ? 'The "' + errorHandler.name + '" validation function' : 'The validation function';
+          return reject(new AssertionError({
+            actual: actualRejectionResult,
+            expected: errorHandler,
+            message: message || validationFunctionName + ' is expected to return "true". Received ' + handlerFuncResult + '\n\nCaught error:\n\n' + actualRejectionResult,
+            operator: stackStartFn.name,
+            stackStartFn: stackStartFn
+          }));
         }
       }
       if (typeof errorHandler === 'object') {
diff --git a/test/test.js b/test/test.js
index 1095aa2..6f390dc 100644
--- a/test/test.js
+++ b/test/test.js
@@ -179,15 +179,32 @@ implementations.forEach(function (impl) {
               assert(nothing === undefined);
             }, shouldNotBeRejected);
           });
-          it('when actual error is NOT an instanceof `<Class>`, rejects with the actual error.', function () {
-            return rejects(
-              willReject(new TypeError('the original error message')),
-              RangeError
-            ).then(shouldNotBeFulfilled, function (err) {
-              assert(err instanceof TypeError);
-              assert.equal(err.message, 'the original error message');
+          // < Node13
+          if (impl.name === 'official implementation' && semver.satisfies(process.version, '< 13.0.0')) {
+            it('when actual error is NOT an instanceof `<Class>`, rejects with the actual error.', function () {
+              return rejects(
+                willReject(new TypeError('the original error message')),
+                RangeError
+              ).then(shouldNotBeFulfilled, function (err) {
+                assert(err instanceof TypeError);
+                assert.equal(err.message, 'the original error message');
+              });
             });
-          });
+          } else {
+            it('when actual error is NOT an instanceof `<Class>`, rejects with AssertionError.', function () {
+              var te = new TypeError('the original error message');
+              return rejects(
+                willReject(te),
+                RangeError
+              ).then(shouldNotBeFulfilled, function (err) {
+                assert(err instanceof assert.AssertionError);
+                assert.equal(err.actual, te);
+                assert.equal(err.expected, RangeError);
+                assert(/The error is expected to be an instance of "RangeError". Received "TypeError"/.test(err.message));
+                assert(/the original error message/.test(err.message));
+              });
+            });
+          }
           describe('works well with ES2015 classes that extends Error', function () {
             class ES2015Error extends Error {
             }
@@ -201,15 +218,32 @@ implementations.forEach(function (impl) {
                 assert(nothing === undefined);
               }, shouldNotBeRejected);
             });
-            it('unmatch case, rejects with the original error.', function () {
-              return rejects(
-                willReject(new AnotherES2015Error('bar')),
-                ES2015Error
-              ).then(shouldNotBeFulfilled, function (err) {
-                assert(err instanceof AnotherES2015Error);
-                assert.equal(err.message, 'bar');
+            // < Node13
+            if (impl.name === 'official implementation' && semver.satisfies(process.version, '< 13.0.0')) {
+              it('unmatch case, rejects with the original error.', function () {
+                return rejects(
+                  willReject(new AnotherES2015Error('bar')),
+                  ES2015Error
+                ).then(shouldNotBeFulfilled, function (err) {
+                  assert(err instanceof AnotherES2015Error);
+                  assert.equal(err.message, 'bar');
+                });
               });
-            });
+            } else {
+              it('unmatch case, rejects with AssertionError.', function () {
+                var another = new AnotherES2015Error('bar');
+                return rejects(
+                  willReject(another),
+                  ES2015Error
+                ).then(shouldNotBeFulfilled, function (err) {
+                  assert(err instanceof assert.AssertionError);
+                  assert.equal(err.actual, another);
+                  assert.equal(err.expected, ES2015Error);
+                  assert(/The error is expected to be an instance of "ES2015Error". Received "AnotherES2015Error"/.test(err.message));
+                  assert(/bar/.test(err.message));
+                });
+              });
+            }
           });
           it('appends `error.name` as expected error class name to the message if the `promiseFn` is not rejected.', function () {
             return rejects(
@@ -232,17 +266,38 @@ implementations.forEach(function (impl) {
               assert(nothing === undefined);
             }, shouldNotBeRejected);
           });
-          it('when returned value of validation function is NOT `true`, rejects with the actual error.', function () {
-            return rejects(
-              willReject(new RangeError('Wrong range')),
-              function (err) {
+          // < Node13
+          if (impl.name === 'official implementation' && semver.satisfies(process.version, '< 13.0.0')) {
+            it('when returned value of validation function is NOT `true`, rejects with the actual error.', function () {
+              return rejects(
+                willReject(new RangeError('Wrong range')),
+                function (err) {
+                  return ((err instanceof TypeError) && /type/.test(err));
+                }
+              ).then(shouldNotBeFulfilled, function (err) {
+                assert(err instanceof RangeError);
+                assert.equal(err.message, 'Wrong range');
+              });
+            });
+          } else {
+            it('when returned value of validation function is NOT `true`, rejects with AssertionError.', function () {
+              var e = new RangeError('Wrong range');
+              const handlerFn = (err) => {
                 return ((err instanceof TypeError) && /type/.test(err));
-              }
-            ).then(shouldNotBeFulfilled, function (err) {
-              assert(err instanceof RangeError);
-              assert.equal(err.message, 'Wrong range');
+              };
+              return rejects(
+                willReject(e),
+                handlerFn
+              ).then(shouldNotBeFulfilled, function (err) {
+                assert(err instanceof assert.AssertionError);
+                assert.equal(err.actual, e);
+                assert.equal(err.expected, handlerFn);
+
+                assert(/The "handlerFn" validation function is expected to return "true". Received false/.test(err.message), `actual [${err.message}]`);
+                assert(/RangeError: Wrong range/.test(err.message), `actual [${err.message}]`);
+              });
             });
-          });
+          }
           it('if Error is thrown from validation function, rejects with the error.', function () {
             var e = new RangeError('the original error message');
             var te = new TypeError('some programming error');