We are motivated by the following problem: which curves are d-sheeted covering spaces of the projective line? We discuss the moduli space of d-sheeted coverings, the Hurwitz Scheme, together with a particular compactification due to Mumford and Harris. This compactification, and its relationship to the compactification of the moduli space of genus g curves, allows us to use graph-theoretic tools to help determine whether certain general curves are d-sheeted coverings of P^1.