A big theme in contemporary number theory is "arithmetic statistics": what does the class group of a random number field look like? What do the zeroes of a random L-function look like? What does a random rational point on a variety look iike? In this talk we will explain how arithmetic statistics problems over function fields are naturally tied to topological questions about stability for homology groups of certain moduli spaces; in particular, we will explain how a stability theorem for Hurwitz spaces (moduli spaces of finite branched covers of the line) can be used to prove a version of the Cohen-Lenstra conjectures over function fields. There will be some overlap with a talk I gave at Duke in December 2009, but many things which were speculations then are theorems now.