How do soundcloud keep music playing on when navigating?
HTML5 has introduced a new API, called HTML5 History API. You can read about using it here – http://diveintohtml5.info/history.html
This API lets you update browser address bar with JavaScript, so you can change the contents of the page according to the URL. The use of AJAX doesn't really have to do with UI changes, you could check the networks tab in Chrome's developer tools to see that requests are issued with XHR.