It is a classical problem to count the number of geometric objects satisfying various conditions. In the 1990s, Kontsevich obtained a beautiful formula for the number of rational plane curves of degree d passing through 3d-1 general points by using Gromov-Witten theory of the projective plane. More recently, orbifold Gromov-Witten theory has been used to study generalizations of the McKay correspondence and crepant resolutions. I will discuss these developments, as well as give an approach to further problems in enumerative geometry using orbifold quantum cohomology.