It is well-known that certain aspects of the Riemann zeta-function are modelled by a randomly chosen element of a large matrix group. In fact, many natural phenomena in number theory are modelled by randomly chosen elements from ”large” finite groups, e.g. permutation groups or large matrix groups over a fixed finite field, or, for that matter, more interesting groups. I would like to discuss several fairly down-to-earth examples of this. The example I will eventually discuss in detail will be related to the Cohen-Lenstra heuristics, but I will explain precisely what this is and how we would like to generalize it. This all represents joint work with Jordan Ellenberg.