A lot has been said about homeomorphism and diffeomorphism groups of spaces (more diffeo), starting with von Neumann and Ulam in the mid 40s (probably before then, as well), and until today. I would advise looking at papers by D.B.A. Epstein in the late sixties, John Mather in the early seventies, and a cool paper by D. Calegari and M. Freedman a couple of years ago (with an even cooler appendix by Y. de Cournouiller). The key fact (discovered very early on) is that these groups are perfect (equal to their commutator subgroup) and simple. It is an interesting question (which seems to be due to myself, but it is hard to believe that von Neumann/Ulam did not already wonder about this) whether every homeomorphism/diffeomorphism is, in fact, a commutator.