Mirror symmetry is a physics duality which allows us to classify symplectic manifolds in terms of associated algebraic geometric data. I will explain the geometric formulation of mirror symmetry for complete intersections in toric varieties and will discuss some recent progress in the general non-complete intersection case. I will describe explicitly the mirror map for del Pezzo surfaces and supporting evidence for the homological mirror correspondence in this case. This is a joint work with Auroux, Katzarkov and Orlov.