diff --git a/types/jquery/JQuery.d.ts b/types/jquery/JQuery.d.ts
index cbcc12d956..ed0147a89f 100644
--- a/types/jquery/JQuery.d.ts
+++ b/types/jquery/JQuery.d.ts
@@ -83,6 +83,7 @@ $( document.body )
      * @since 1.4
      */
     add(selector: JQuery.Selector, context: Element): this;
+    // TODO: The return type should reflect newly selected types.
     /**
      * Create a new jQuery object with elements added to the set of matched elements.
      * @param selector_elements_html_selection _@param_ `selector_elements_html_selection`
@@ -226,7 +227,7 @@ collection.css( "background", "yellow" );
 </html>
 ```
      */
-    add(selector_elements_html_selection: JQuery.Selector | JQuery.TypeOrArray<Element> | JQuery.htmlString | JQuery): this;
+    add(selector_elements_html_selection: JQuery.Selector | JQuery.TypeOrArray<Element> | JQuery.htmlString | JQuery | JQuery.Node): this;
     /**
      * Add the previous set of elements on the stack to the current set, optionally filtered by a selector.
      * @param selector A string containing a selector expression to match the current set of elements against.
@@ -5141,7 +5142,9 @@ $( "div b" )
 </html>
 ```
      */
-    html(htmlString_function: JQuery.htmlString | ((this: TElement, index: number, oldhtml: JQuery.htmlString) => JQuery.htmlString)): this;
+    html(htmlString_function: JQuery.htmlString |
+                              JQuery.Node |
+                              ((this: TElement, index: number, oldhtml: JQuery.htmlString) => JQuery.htmlString | JQuery.Node)): this;
     /**
      * Get the HTML contents of the first element in the set of matched elements.
      * @see \`{@link https://api.jquery.com/html/ }\`
@@ -9990,7 +9993,14 @@ $( "button" ).on( "click", function() {
 </html>
 ```
      */
-    replaceWith(newContent_function: JQuery.htmlString | JQuery | JQuery.TypeOrArray<Element> | ((this: TElement) => any)): this;
+    replaceWith(newContent_function: JQuery.htmlString |
+                                     JQuery<JQuery.Node> |
+                                     JQuery.TypeOrArray<Element> |
+                                     JQuery.Node |
+                                     ((this: TElement, index: number, oldhtml: JQuery.htmlString) => JQuery.htmlString |
+                                                                                                     JQuery<JQuery.Node> |
+                                                                                                     JQuery.TypeOrArray<Element> |
+                                                                                                     JQuery.Node)): this;
     /**
      * Bind an event handler to the "resize" JavaScript event, or trigger that event on an element.
      * @param eventData An object containing data that will be passed to the event handler.
diff --git a/types/jquery/jquery-tests.ts b/types/jquery/jquery-tests.ts
index 9e4dea80ee..0acf6e0d24 100644
--- a/types/jquery/jquery-tests.ts
+++ b/types/jquery/jquery-tests.ts
@@ -5565,11 +5565,21 @@ function JQuery() {
             $('p').replaceWith([new HTMLElement()]);
 
             // $ExpectType JQuery<HTMLElement>
-            $('p').replaceWith(function() {
+            $('p').replaceWith(document.createTextNode('bar'));
+
+            // $ExpectType JQuery<HTMLElement>
+            $('p').replaceWith(document.createComment('bar'));
+
+            // $ExpectType JQuery<HTMLElement>
+            $('p').replaceWith(function(index, oldhtml) {
                 // $ExpectType HTMLElement
                 this;
+                // $ExpectType number
+                index;
+                // $ExpectType string
+                oldhtml;
 
-                return this;
+                return undefined! as JQuery.htmlString | JQuery<JQuery.Node> | JQuery.TypeOrArray<Element> | JQuery.Node;
             });
         }
 
@@ -5693,6 +5703,12 @@ function JQuery() {
             // $ExpectType JQuery<HTMLElement>
             $('p').html('<span></span>');
 
+            // $ExpectType JQuery<HTMLElement>
+            $('p').html(document.createTextNode('bar'));
+
+            // $ExpectType JQuery<HTMLElement>
+            $('p').html(document.createComment('bar'));
+
             // $ExpectType JQuery<HTMLElement>
             $('p').html(function(index, oldhtml) {
                 // $ExpectType HTMLElement
@@ -5702,7 +5718,12 @@ function JQuery() {
                 // $ExpectType string
                 oldhtml;
 
-                return oldhtml;
+                switch (index) {
+                    case 0: return document.createTextNode('bar');
+                    case 1: return document.createComment('bar');
+
+                    default: return oldhtml;
+                }
             });
 
             // $ExpectType string
@@ -5805,6 +5826,12 @@ function JQuery() {
 
             // $ExpectType JQuery<HTMLElement>
             $('p').add($('span'));
+
+            // $ExpectType JQuery<HTMLElement>
+            $('p').add(document.createTextNode('bar'));
+
+            // $ExpectType JQuery<HTMLElement>
+            $('p').add(document.createComment('bar'));
         }
 
         function closest() {