Mirror Symmetry: an overview

The mirror principle of string theory relates the symplectic topology of a Kahler manifold (counts of complex curves) to the differential geometry of its mirror family. Much of the current excitement about mirror symmetry stems from the 1991 paper of Candelas, de la Ossa, Green, and Parkes, which gave an explicit prediction for counts of rational curves in the quintic threefold. This has been followed by many predictions (and occasional verifications) for other types of curve counts. In this talk, I will touch on various flavors of mirror symmetry ("classical", real, and homological) and indicate the current state of affairs in the subject.