Mirror symmetry pertains to string theory in the presence of branes. In this setting, Gromov-Witten invariants involve pseudoholomorphic maps from Riemann surfaces with boundary into a symplectic manifold, the boundary mapping into a Lagrangian submanifold (brane). These invariants can be predicted by a mirror map. I will verify these predictions, as well as some integrality conjectures, through a non-rigorous localization calculation. This calculation can be related to a rigorously defined closed-string calculation, in which case mirror symmetry can be "proved."