Mirror symmetry is a conjectural duality between pairs of Calabi-Yau manifolds with additional structures. We explain two instances of mirror symmetry. First, the number of rational curves on a quintic 3-fold can be obtained from the complex periods on the mirror manifold. Second, coherent sheaves on an elliptic curve are closely related to closed geodesics on a torus. Based on these observations, we formulate homological mirror symmetry as an equivalence of categories.