The classical Schur-Horn theorem characterizes the set of diagonals of the unitary orbit of a self-adjoint matrix in terms of a set of linear inequalities called majorization. In 2002 Kadison discovered a characterization of the diagonals of orthogonal projections on infinite dimensional Hilbert spaces. Since Kadison's breakthrough there has been a great deal of work by several authors to extend the Schur-Horn theorem to all self-adjoint operators on infinite dimensional Hilbert spaces. In this talk we discuss our recent contributions to this effort. In particular we present a characterization of the diagonals of self-adjoint operators with finite spectrum.