Brill-Noether theory studies the maps of curves C to projective spaces. The classical Brill-Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz-Brill-Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. I'll also present recent results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane.