Let G be a finite group and c a conjugacy class of G. Hurwitz spaces are certain spaces built out of the unordered configuration space of the plane and (G, c). I will talk about work of Ellenberg—Venkatesh—Westerland, where they prove that the homology of these spaces stabilize in certain cases. Time permitting, I will also talk about how they apply this stability result to say something about a heuristic in number theory about class groups of quadratic field extensions over the field F_q(t), with q=p^n for p an odd prime.