Abstract: We describe a new homological stability result for a generalized version of Hurwitz spaces. This work, joint with Jordan Ellenberg, builds on previous work of Ellenberg-Venkatesh-Westerland, but generalizes it in two directions. First, we work with covers of arbitrary punctured Riemann surfaces instead of just the disc. Second, our result more generally applies not just to Hurwitz spaces, but also “coefficient systems,” which are essentially a sequence of compatible local systems on configurations spaces. We will then explain how this homological stability result is employed to prove conjectures from number theory relating to the distributions of ranks of elliptic curves.