The equations for Abelian Higgs vortices (magnetic flux vortices) on a plane or a more general surface are generally not integrable, but for vortices on a hyperbolic plane of curvature -1/2 they are. This talk will present (almost explicit) vortex solutions on certain compact hyperbolic surfaces. Also to be discussed are two asymptotically solvable problems for vortices: the effective vortex motion on a large surface with small curvature, and the structure of vortex solutions on a small surface where the vortices are about to dissolve (and the equations linearize). These results (obtained with N. Rink and with N. Romao) bring vortex theory closer to classical results on the complex and metric geometry of Riemann surfaces.