Let \(G\) be a permutation group on a set \(X\). The authors consider the ordinary generating functions \(f_G(t)\) (for the number of orbits of \(G\) on subsets of size \(n\)), and the exponential generating functions \(F_G(t)\) (for the number of orbits on \(n\)-tuples of distinct elements) and \(F_G^*(t)\) (for the number of orbits on all \(n\)-tuples of elements). The last two are related by the identity \(F_G^*(t)=F_G(e^t-1)\). They show how to compute these functions in various cases, and explain their relationship with other combinatorial objects. In particular, they show how to compute \(F_G^*(t)\) for the product action of the direct product of two permutation groups, and for the product action of a wreath product.