Let G be an undirected graph on n vertices, and let S(G) denote the set of all n-by-n symmetric matrices whose graph is G; the diagonal entries ofA in (G) are free. After reviewing prior results, we will discuss recent results thataddress the question of what lists of multiplicities for the eigenvalues may occur among matrices inS(G). This represents joint work of A. Leal-Duarte (Coimbra), Carlos Saiago (Lisbon),and two recent REU students, Biran Sutton and Andrew Witt.