Microsoft has come a long way in supporting other browsers, meaning browsers other than Internet Explorer. During SharePoint Conference 2012 in Las Vegas last November, several people made public statements saying that all major browsers (i.e., Firefox, Chrome and Safari) are now supported across the platform. While Microsoft has made massive progress in cross browser support those statements unfortunately are incorrect – we still need Internet Explorer for certain tasks.
Next question: well, what tasks?
Fortunately not only cross browser support has been improved, but also information on what IE actually is still needed for. Microsoft has published an article to Plan browser support in SharePoint 2013. This article contains a Browser Support Matrix as well as the information you are actually looking for: a list of areas & functions in SharePoint 2013 where ActiveX Controls are still in use. This list reads as “This is what you need Internet Explorer for”.