| Brill-Noether theory of curves on K3 surfaces is well understood. Quite little is known for curves lying on abelian surfaces. Given a general abelian surface S with polarization L of type (1,n), we will first show that a general curve in |L| is Brill-Noether general. We will then study the locus |L|^r_d of smooth curves in |L| possessing a g^r_d and prove that this is nonempty in some unexpected cases (with negative Brill-Noether number). As an application, we obtain the existence of a component of the Brill-Noether locus M^r_{g,d} having the expected codimension in the moduli space of curves M_g. Time permitting, we will mention applications to enumerative geometry and hyperkähler manifolds. Most of this work is joint with A. L. Knutsen and G. Mongardi. |