The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether Theorem, which says that the space of linear systems with given discrete invariants on a general curve has the expected dimension. In this talk, we will discuss a generalization of this theorem to curves that are general in the Hurwitz space, rather than in the moduli space of curves. This is joint work with Kaelin Cook-Powell.