I will discuss some models of what a "random abelian group" is, and some conjectures (the Cohen-Lenstra heuristics of the title) about how they show up in number theory. I'll then discuss the function field setting and a proof of these heuristics, with Ellenberg and Westerland. The proof is an example of a link between analytic number theory and certain classes of results in algebraic topology ("homological stability").