This talk will focus on recent work concerning the geometry of the mapping class group (alias Teichmueller modular groups). One of the main results I'll discuss is that, up to finite groups, any finitely generated group quasi-isometric to a mapping class group is a mapping class group. I intend to make this talk accessible to a broad audience of geometer/topologists. This is joint work with Bruce Kleiner, Yair Minsky, and Lee Mosher.