I will start by explaining what mirror symmetry is about, paying special attention to the mirror map which matches up the family of symplectic forms on one manifold with the family of complex structures on another. I will explain how this works for Batyrev's beautiful toric construction of mirror families from dual reflexive polytopes. Then I will give a template for proving cases of Kontsevich's homological mirror symmetry conjecture, based on a versality result for the Fukaya category, which roughly gives a criterion for the existence of a mirror map. The proof can be completed when the reflexive polytope in Batyrev's construction is a simplex: this special case of the construction is due to Greene and Plesser. The latter result is joint work with Ivan Smith.