Classical Brill-Noether theory is the study of the geometry of projective curves, and it is defined in terms of the cohomological properties of line bundles. These line bundles vary in the Picard space of the curve, and the geometry of that space determines much of the geometry of the embedding. In this talk I will explore the generalizations of this study to higher rank vector bundles and their moduli spaces, where we replace the base curve by the projective plane. The properties of Brill-Noether loci inside moduli spaces of sheaves on surfaces are largely unknown, but in recent joint work with Yeqin Liu and Woohyung Lee, we have established many of their fundamental properties on the projective plane.