Abstract: The goal of this talk will be to give a mathematical overview of certain aspects of mirror symmetry. Mirror symmetry is a duality between the complex geometry of one manifold and the symplectic geometry of another. In some respects, this duality reflects the similarity in the definitions of complex and symplectic structures. However, mirror symmetry also predicts a relationship between certain problems of strikingly different nature. For example, the problem of calculating period integrals for a complex manifold is linear. However, the mirror problem of counting holomorphic maps from the Riemann sphere to a symplectic manifold is highly non-linear. Recent work in symplectic geometry has led to a new example of mirror symmetry in the case when the Riemann sphere is replaced by the disk. This talk will focus on explicit examples.