This is a joint work with P.Deift. We give a rigorous proof of the Universality Conjecture in Random Matrix Theory for orthogonal (beta=1) and symplectic (beta=4) ensembles in the scaling limit for a class of polynomial potentials whose equilibrium measure is supported on a single interval. Our starting point is Widom's representation of the correlation kernels for the beta=1,4 cases in terms of the unitary (beta=2) correlation kernel plus a correction. In the asymptotic analysis of the correction terms we use amongst other things differential equations for the derivatives of orthogonal polynomials (OP's) due to Tracy-Widom, and uniform Plancherel-Rotach type asymptotics for OP's due to Deift-Kriecherbauer-McLaughlin-Venakides-Zhou. The problem reduces to a small norm problem for a certain matrix of a fixed size that is equal to the degree of the polynomial potential.