Abstract: I'll discuss (orbifold) symmetric products of projective spaces Sym^d(P^r), focusing on the structure of orbits of the natural torus action. These orbits vary in nice toric moduli spaces -- for example, components of the moduli space of 1-dimensional orbits are naturally identified with certain toric compactifications of M_{0,n}. I will then discuss how one can use these orbits to calculate Gromov-Witten invariants of Sym^d(P^r). If I have time, I'll also talk about how this story works when Sym^d(P^r) is replaced with the Hilbert scheme of points Hilb^d(P^r).