Through 3 general points and 6 general lines in P^3, there are exactly 190 twisted cubics; 190 is a Gromov-Witten invariant of P^3. I will introduce Gromov-Witten invariants of a smooth complex projective variety X, and show how an equivariant structure on a X can help us compute its Gromov-Witten invariants. In the case when X is a toric variety, Kontsevich used this method to compute any Gromov-Witten invariant of X. Givental and Lian-Liu-Yau used Kontsevich's strategy to prove a mirror theorem, which states that Gromov-Witten invariants of X have an interesting rigid structure predicted by physicists. I will discuss the difficulties that arise when X is not toric. The subject of my thesis is the nontoric orbifold X=Sym^d(P^r), the symmetric product of projective space. By studying the equivariant geometry of Sym^d(P^r) I extend the results of Kontsevich to Sym^d(P^r), and prove a mirror theorem for Sym^d(P^r). Speaker(s): Robert Silversmith (University of Michigan)