diff --git a/Gruntfile.js b/Gruntfile.js
index 904abbcea..7fa324399 100644
--- a/Gruntfile.js
+++ b/Gruntfile.js
@@ -14,7 +14,15 @@ module.exports = function( grunt ) {
 
 	var fs = require( "fs" ),
 		gzip = require( "gzip-js" ),
-		isTravis = process.env.TRAVIS;
+		isTravis = process.env.TRAVIS,
+		oldNode = /^v6\./.test( process.version );
+
+	// Support: Node.js <8
+	// Skip running tasks that dropped support for Node.js 6
+	// in those Node versions.
+	function runIfNewNode( task ) {
+		return oldNode ? "print_old_node_message:" + task : task;
+	}
 
 	if ( !grunt.option( "filename" ) ) {
 		grunt.option( "filename", "jquery.js" );
@@ -220,6 +228,33 @@ module.exports = function( grunt ) {
 				browsers: [ isTravis ? "ChromeHeadlessNoSandbox" : "ChromeHeadless" ]
 			},
 
+			jsdom: {
+				options: {
+					files: [
+						"test/data/jquery-1.9.1.js",
+						"test/data/testinit-jsdom.js",
+
+						// We don't support various loading methods like AMD,
+						// choosing a version etc. for jsdom.
+						"dist/jquery.js",
+
+						// Replacement for testinit.js#loadTests()
+						"test/data/testrunner.js",
+
+						// jsdom only runs basic tests
+						"test/unit/basic.js",
+
+						{ pattern: "external/**", included: false, served: true },
+						{
+							pattern: "test/**/*.@(js|css|jpg|html|xml|svg)",
+							included: false,
+							served: true
+						}
+					]
+				},
+				browsers: [ "jsdom" ]
+			},
+
 			// To debug tests with Karma:
 			// 1. Run 'grunt karma:chrome-debug' or 'grunt karma:firefox-debug'
 			//    (any karma subtask that has singleRun=false)
@@ -285,6 +320,13 @@ module.exports = function( grunt ) {
 	// Integrate jQuery specific tasks
 	grunt.loadTasks( "build/tasks" );
 
+	// Support: Node.js <8
+	// Print a message on Node.js <8 notifying the task is skipped there.
+	grunt.registerTask( "print_old_node_message", function() {
+		var task = [].slice.call( arguments ).join( ":" );
+		grunt.log.writeln( "Old Node.js detected, running the task \"" + task + "\" skipped..." );
+	} );
+
 	grunt.registerTask( "lint", [
 		"jsonlint",
 
@@ -305,7 +347,13 @@ module.exports = function( grunt ) {
 	] );
 
 	grunt.registerTask( "test:fast", "node_smoke_tests" );
-	grunt.registerTask( "test:slow", "promises_aplus_tests" );
+	grunt.registerTask( "test:slow", [
+		"promises_aplus_tests",
+
+		// Support: Node.js <8
+		// Karma no longer supports Node.js <8 as it relies on async-await internally.
+		runIfNewNode( "karma:jsdom" )
+	] );
 
 	grunt.registerTask( "test", [
 		"test:fast",
diff --git a/package.json b/package.json
index ffe9d1b1d..4124d2d93 100644
--- a/package.json
+++ b/package.json
@@ -51,6 +51,7 @@
     "karma-chrome-launcher": "2.2.0",
     "karma-firefox-launcher": "1.1.0",
     "karma-ie-launcher": "1.0.0",
+    "karma-jsdom-launcher": "7.1.0",
     "karma-qunit": "3.0.0",
     "load-grunt-tasks": "4.0.0",
     "native-promise-only": "0.8.1",
diff --git a/test/.eslintrc.json b/test/.eslintrc.json
index 2c724626f..a23ea5f98 100644
--- a/test/.eslintrc.json
+++ b/test/.eslintrc.json
@@ -55,7 +55,10 @@
 
 	"overrides": [
 		{
-			"files": ["data/core/jquery-iterability-transpiled-es6.js"],
+			"files": [
+				"data/core/jquery-iterability-transpiled-es6.js",
+				"data/testinit-jsdom.js"
+			],
 			"parserOptions": {
 				"ecmaVersion": 2015
 			}
diff --git a/test/data/testinit-jsdom.js b/test/data/testinit-jsdom.js
new file mode 100644
index 000000000..e0830cc92
--- /dev/null
+++ b/test/data/testinit-jsdom.js
@@ -0,0 +1,40 @@
+"use strict";
+
+// Support: jsdom 13.2+
+// jsdom implements a throwing `window.scrollTo`.
+QUnit.config.scrolltop = false;
+
+const FILEPATH = "/test/data/testinit-jsdom.js";
+const activeScript = document.currentScript;
+const parentUrl = activeScript && activeScript.src ?
+	activeScript.src.replace( /[?#].*/, "" ) + FILEPATH.replace( /[^/]+/g, ".." ) + "/" :
+	"../";
+const supportjQuery = this.jQuery;
+
+// baseURL is intentionally set to "data/" instead of "".
+// This is not just for convenience (since most files are in data/)
+// but also to ensure that urls without prefix fail.
+// Otherwise it's easy to write tests that pass on test/index.html
+// but fail in Karma runner (where the baseURL is different).
+const baseURL = parentUrl + "test/data/";
+
+// Setup global variables before loading jQuery for testing .noConflict()
+supportjQuery.noConflict( true );
+window.originaljQuery = this.jQuery = undefined;
+window.original$ = this.$ = "replaced";
+
+/**
+ * Add random number to url to stop caching
+ *
+ * Also prefixes with baseURL automatically.
+ *
+ * @example url("index.html")
+ * @result "data/index.html?10538358428943"
+ *
+ * @example url("mock.php?foo=bar")
+ * @result "data/mock.php?foo=bar&10538358345554"
+ */
+function url( value ) {
+	return baseURL + value + ( /\?/.test( value ) ? "&" : "?" ) +
+		new Date().getTime() + "" + parseInt( Math.random() * 100000, 10 );
+}
diff --git a/test/unit/basic.js b/test/unit/basic.js
index b57c0c0ba..bed27dcba 100644
--- a/test/unit/basic.js
+++ b/test/unit/basic.js
@@ -193,7 +193,9 @@ QUnit.test( "manipulation", function( assert ) {
 	);
 } );
 
-QUnit.test( "offset", function( assert ) {
+// Support: jsdom 13.2+
+// jsdom returns 0 for offset-related properties
+QUnit[ /jsdom\//.test( navigator.userAgent ) ? "skip" : "test" ]( "offset", function( assert ) {
 	assert.expect( 3 );
 
 	var parent = jQuery( "<div style='position:fixed;top:20px;'/>" ).appendTo( "#qunit-fixture" ),