Hurwitz spaces, certain finite covers of unordered configuration spaces of the plane, and their generalizations play a fundamental role in the recent flurry of remarkable work connecting topology and number theory, such as Ellenberg-Venkatesh-Westerland, Ellenberg-Tran-Westerland, Liu-Wood-Zureick-Brown, and Ellenberg-Landesman. In this talk, we will explain the approach of Ellenberg-Tran-Westerland to produce an asymptotic bound on the homology of Hurwitz spaces, starting with integrating the classical Fox-Neuwirth stratification of configuration spaces into the twisted setting. Time permitting, we will explore ways to expand this framework to study the homology of generalized Hurwitz spaces over an arbitrary curve, which has potential applications in number theory.