Solution 1:

Ultimate approach so far, with smallest code:

(IE11+, FF, Webkit)

Using MutationObserver and falling back to the deprecated Mutation events if needed:
(Example below if only for DOM changes concerning nodes appended or removed)

var observeDOM = (function(){
  var MutationObserver = window.MutationObserver || window.WebKitMutationObserver;

  return function( obj, callback ){
    if( !obj || obj.nodeType !== 1 ) return; 

    if( MutationObserver ){
      // define a new observer
      var mutationObserver = new MutationObserver(callback)

      // have the observer observe foo for changes in children
      mutationObserver.observe( obj, { childList:true, subtree:true })
      return mutationObserver
    }
    
    // browser support fallback
    else if( window.addEventListener ){
      obj.addEventListener('DOMNodeInserted', callback, false)
      obj.addEventListener('DOMNodeRemoved', callback, false)
    }
  }
})()


//------------< DEMO BELOW >----------------

// add item
var itemHTML = "<li><button>list item (click to delete)</button></li>",
    listElm = document.querySelector('ol');

document.querySelector('body > button').onclick = function(e){
  listElm.insertAdjacentHTML("beforeend", itemHTML);
}

// delete item
listElm.onclick = function(e){
  if( e.target.nodeName == "BUTTON" )
    e.target.parentNode.parentNode.removeChild(e.target.parentNode);
}
    
// Observe a specific DOM element:
observeDOM( listElm, function(m){ 
   var addedNodes = [], removedNodes = [];

   m.forEach(record => record.addedNodes.length & addedNodes.push(...record.addedNodes))
   
   m.forEach(record => record.removedNodes.length & removedNodes.push(...record.removedNodes))

  console.clear();
  console.log('Added:', addedNodes, 'Removed:', removedNodes);
});


// Insert 3 DOM nodes at once after 3 seconds
setTimeout(function(){
   listElm.removeChild(listElm.lastElementChild);
   listElm.insertAdjacentHTML("beforeend", Array(4).join(itemHTML));
}, 3000);
<button>Add Item</button>
<ol>
  <li><button>list item (click to delete)</button></li>
  <li><button>list item (click to delete)</button></li>
  <li><button>list item (click to delete)</button></li>
  <li><button>list item (click to delete)</button></li>
  <li><em>&hellip;More will be added after 3 seconds&hellip;</em></li>
</ol>

Solution 2:

2015 update, new MutationObserver is supported by modern browsers:

Chrome 18+, Firefox 14+, IE 11+, Safari 6+

If you need to support older ones, you may try to fall back to other approaches like the ones mentioned in this 5 (!) year old answer below. There be dragons. Enjoy :)


Someone else is changing the document? Because if you have full control over the changes you just need to create your own domChanged API - with a function or custom event - and trigger/call it everywhere you modify things.

The DOM Level-2 has Mutation event types, but older version of IE don't support it. Note that the mutation events are deprecated in the DOM3 Events spec and have a performance penalty.

You can try to emulate mutation event with onpropertychange in IE (and fall back to the brute-force approach if non of them is available).

For a full domChange an interval could be an over-kill. Imagine that you need to store the current state of the whole document, and examine every element's every property to be the same.

Maybe if you're only interested in the elements and their order (as you mentioned in your question), a getElementsByTagName("*") can work. This will fire automatically if you add an element, remove an element, replace elements or change the structure of the document.

I wrote a proof of concept:

(function (window) {
    var last = +new Date();
    var delay = 100; // default delay

    // Manage event queue
    var stack = [];

    function callback() {
        var now = +new Date();
        if (now - last > delay) {
            for (var i = 0; i < stack.length; i++) {
                stack[i]();
            }
            last = now;
        }
    }

    // Public interface
    var onDomChange = function (fn, newdelay) {
        if (newdelay) delay = newdelay;
        stack.push(fn);
    };

    // Naive approach for compatibility
    function naive() {

        var last = document.getElementsByTagName('*');
        var lastlen = last.length;
        var timer = setTimeout(function check() {

            // get current state of the document
            var current = document.getElementsByTagName('*');
            var len = current.length;

            // if the length is different
            // it's fairly obvious
            if (len != lastlen) {
                // just make sure the loop finishes early
                last = [];
            }

            // go check every element in order
            for (var i = 0; i < len; i++) {
                if (current[i] !== last[i]) {
                    callback();
                    last = current;
                    lastlen = len;
                    break;
                }
            }

            // over, and over, and over again
            setTimeout(check, delay);

        }, delay);
    }

    //
    //  Check for mutation events support
    //

    var support = {};

    var el = document.documentElement;
    var remain = 3;

    // callback for the tests
    function decide() {
        if (support.DOMNodeInserted) {
            window.addEventListener("DOMContentLoaded", function () {
                if (support.DOMSubtreeModified) { // for FF 3+, Chrome
                    el.addEventListener('DOMSubtreeModified', callback, false);
                } else { // for FF 2, Safari, Opera 9.6+
                    el.addEventListener('DOMNodeInserted', callback, false);
                    el.addEventListener('DOMNodeRemoved', callback, false);
                }
            }, false);
        } else if (document.onpropertychange) { // for IE 5.5+
            document.onpropertychange = callback;
        } else { // fallback
            naive();
        }
    }

    // checks a particular event
    function test(event) {
        el.addEventListener(event, function fn() {
            support[event] = true;
            el.removeEventListener(event, fn, false);
            if (--remain === 0) decide();
        }, false);
    }

    // attach test events
    if (window.addEventListener) {
        test('DOMSubtreeModified');
        test('DOMNodeInserted');
        test('DOMNodeRemoved');
    } else {
        decide();
    }

    // do the dummy test
    var dummy = document.createElement("div");
    el.appendChild(dummy);
    el.removeChild(dummy);

    // expose
    window.onDomChange = onDomChange;
})(window);

Usage:

onDomChange(function(){ 
    alert("The Times They Are a-Changin'");
});

This works on IE 5.5+, FF 2+, Chrome, Safari 3+ and Opera 9.6+

Solution 3:

The following example was adapted from Mozilla Hacks' blog post and is using MutationObserver.

// Select the node that will be observed for mutations
var targetNode = document.getElementById('some-id');

// Options for the observer (which mutations to observe)
var config = { attributes: true, childList: true };

// Callback function to execute when mutations are observed
var callback = function(mutationsList) {
    for(var mutation of mutationsList) {
        if (mutation.type == 'childList') {
            console.log('A child node has been added or removed.');
        }
        else if (mutation.type == 'attributes') {
            console.log('The ' + mutation.attributeName + ' attribute was modified.');
        }
    }
};

// Create an observer instance linked to the callback function
var observer = new MutationObserver(callback);

// Start observing the target node for configured mutations
observer.observe(targetNode, config);

// Later, you can stop observing
observer.disconnect();

Browser support: Chrome 18+, Firefox 14+, IE 11+, Safari 6+