Abstract: Mirror symmetry is a correspondence between symplectic geometry on a manifold M and complex geometry on a mirror manifold W. The question of why one sort of geometry on M should be reflected in another sort of geometry on the topologically distinct manifold W, and the question of how to find W given M, are a priori highly mysterious. One attempt to explain the mysteries of mirror symmetry is the SYZ conjecture, which asserts that the mirror manifold W can be realized as the moduli space of certain objects of a category associated to M. In this talk, I will explain how to prove that certain objects of the category associated to M indeed behave like points of W.