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 general, we expect the monodromy of a family to be "big", i.e. as large as possible subject to any geometrical or algebraic constraints arising from the family. In this talk I'll discuss the monodromy of Hurwitz spaces of G-covers, moduli spaces for branched covers of the projective line with Galois group G. I show that if G is center-free and has trivial Schur multiplier the mod $\ell$ monodromy will be big in an appropriate sense if the number of branch points is chosen to be sufficiently large.