Let n >= 4 be an even integer and let M_n be the moduli space of n points on P^1 modulo the action of PGL(2), thought of as the GIT quotient (P^1)^n/PGL(2). The space M_n has a natural projective embedding; let R_n be its projective coordinate ring. The ring R_n was studied classically in the context of invariant theory. In the late nineteenth century, Kempe proved that R_n is generated by its degree one piece. Since that time however, generators for the ideal of relations have not been determined. I will talk about my recent work with Ben Howard, John Millson and Ravi Vakil on understanding generators of this ideal. Our main result is that when n is not six the ideal is generated by a single quadric equation up to symmetry. The n=6 and n=8 cases have a number of special features which I will also mention.