This talk will focus on the use of sum rules to give necessary and sufficient conditions on a spectral measure for Jacobi Matrices, Orthogonal Polynomials on the Unit Circle or Schrodinger Operators to have "potentials" which are of some specific type, for example, in L^2. I'll start with the historically first case of OPUC and Verblunsky's form of Szego's theorem and then focus on my more recent work with Killip.