Abstract: The moduli space $\overline{M}_{g,n}(X,\beta)$ parameterizes maps from genus g curves with n marked points into X of degree \beta in H_2(X). Certain intersection numbers on this space give the famous Gromov-Witten invariants of X. In this talk, I will describe work in progress on understanding the entire Chow/cohomology ring of the space \overline{M}_{0,n}(P^r,d) in terms of additive generators and relations.