In this talk I will discuss some variations on the classical Thompson's groups F, T and V, some geometries on which they act, and finiteness properties of the groups. This builds off of classical work due in part to Brown and Stein. Examples of groups I will discuss include braided and higher dimensional Thompson's groups, due to Brin, and also some new examples being developed under a general program, joint with Witzel. Work on the finiteness properties of these groups is joint with Bux, Fluch, Marschler (nee Schwandt) and Witzel.