It is well-known that permutations of n contain, on average, 1/k cycles of length k, for k = 1, 2, ..., n, and that the total number of cycles of a random permutation of n is normally distributed with mean and variance asymptotic to log n. Furthermore, in the limit of large n, the joint distribution of the number of cycles of each length is a joint Poisson distribution. Let S be a set of positive integers, and consider permutations with all their cycle lengths in S. Choose one of these uniformly at random. What can be said about the number of cycles of each length? In the case where S is finite, almost all cycles are of length max S; for example, the average involution (permutation which squares to the identity) of [n] has about sqrt(n) fixed points. When S is the complement of a finite set (a generalization of derangements) the situation is similar to when S is all the integers. If S has asymptotic density strictly between 0 and 1 -- for example if S is the odd integers -- more interesting phenomena occur, analogous in many ways to unrestricted permutations but with some surprising new twists. I will also give generalizations of these results to ``weighted permutations''. Here each cycle carries a weight, the weight of a permutation is the product of the weights of its cycles, and we choose permutations randomly according to their weights. This is a richer family of objects that encompasses other classical combinatorial problems. The principal tools I use come from singularity analysis of generating functions. I will attempt to give some probabilistic intuition for why permutations look as they do, but no knowledge of probability is assumed.