We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space, which extend those known previously for Laplacians and Schrodinger operators, freeing them from restrictive assumptions on the nature of the spectrum and allowing operators of much more general form. In particular, we allow for the presence of continuous spectrum, which is a necessary underpinning for new proofs of Lieb-Thirring inequalities. We both sharpen and extend universal bounds on spectral gaps and moments of eigenvalues {lambda(k)} of familiar types, and in addition we produce novel kinds of inequalities that are new even in the model cases. These include a family of differential inequalities for generalized Riesz means and a theorem stating that arithmetic means of {lambda(p)(k)}(k=1)(n) with p <= 3 for eigenvalues of Dirichlet Laplacians are universally bounded from above by multiples of the geometric mean, (Pi(n)(k=1) lambda(k))(1/n). For Schrodinger operators and Dirichlet Laplacians, these bounds are Weyl-sharp, i.e., saturated by the standard semiclassical estimates as n -> 8.