**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.