Hurwitz spaces are certain (finite) covers of configuration spaces of points on a line associated to a (finite) group G. In topology, one is interested in the monodromy (group) of these covers, and in computing the homology of Hurwitz spaces. Such problems can also be stated in terms of the fundamental groups of these configuration space - the braid groups. In geometry, one views Hurwitz spaces as moduli spaces of ramified G-covers of a line. An understanding of the set of such covers defined over global fields, or over finite fields, has implications to the qualitative and quantitative inverse Galois problem. It turns out that topological information about Hurwitz spaces helps make progress toward such arithmetic problems. We will survey this landscape emphasizing the more elementary open problems and approaches.