How do I modify the URL without reloading the page?

This can now be done in Chrome, Safari, Firefox 4+, and Internet Explorer 10pp4+!

See this question’s answer for more information: Updating address bar with new URL without hash or reloading the page

Example:

 function processAjaxData(response, urlPath){
     document.getElementById("content").innerHTML = response.html;
     document.title = response.pageTitle;
     window.history.pushState({"html":response.html,"pageTitle":response.pageTitle},"", urlPath);
 }

You can then use window.onpopstate to detect the back/forward button navigation:

window.onpopstate = function(e){
    if(e.state){
        document.getElementById("content").innerHTML = e.state.html;
        document.title = e.state.pageTitle;
    }
};

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For a more in-depth look at manipulating browser history, see this MDN article.