Abstract: A Jacobi matrix is a tridiagonal self-adjoint real matrix with b_n on the main diagonal and a_n on the next two diagonals. The sum rules are a family of trace class formulas relating a_n,b_n and the spectral measure of the matrix. For the latter, eigenvalues outside the essential spectrum and certain Szego-type integrals involving the a.c. part enter. We build on a recent paper of Killip-Simon and extend their sum rules to all Jacobi matrices which are compact perturbations of the free matrix J_0 (with a_n=1 and b_n=0). This is joint work with B. Simon.