Motivated by questions of pattern avoidance on permutations, I consider the number of permutations of [n] with all cycle lengths lying in a finite set S, with m = max S. For involutions (S = {1, 2}) this number is sqrt(n!) times a subexponential factor. I then consider specifically the composition of two involutions on [n] chosen uniformly at random. Statistics of the cycle structure of such involutions can be found using generating functions arising from graph theory, and can be motivated using some probabilistic heuristics. Finally I will consider the number of factorizations of a permutation pi into involutions (that is, ordered pairs (sigma, tau) with pi = tau * sigma). There is a simple formula for the number of factorizations which depends on the cycle type of the permutation, and the number of such factorizations of a random permutation appears to be lognormally distributed.