Elena Fuchs University of Illinois at Urbana-Champaign Thin groups, or groups that are infinite index in their Zariski closure, have become important objects in the context of number theory. We present some rich properties of certain thin groups, and delve into two kinds of questions about thin groups in general. One is how to determine whether a given group is thin, and the other is whether thin groups are in some sense generic. Joint work in part with Meiri, Rivin, and Sarnak.