The monodromy of a family of varieties is a measure of how homology classes vary. Surprisingly, many familiar ideas in number theory, such as Galois representations and Cohen-Lenstra heuristics, are closely linked to monodromy of specific families. In this talk I will define monodromy, and describe original work of computing monodromy for moduli spaces of covers of the projective line (Hurwitz spaces). This work generalizes previous results of Achter-Pries, Yu and Hall on hyperelliptic families. Only basic knowledge of algebraic topology and number theory is required.
