The distributions of eigenvalues of random matrices display universal behavior in two ways. On the one hand these distributions appear in many areas of mathematics and physics including such fields as number theory and combinatorics which have no obvious connection to random matrices. On the other hand these distributions are universal in the sense that there are large classes of ensembles displaying the same distributions in the limit of large matrix dimensions. In this talk both aspects of universality will be briefly surveyed. We will also present recent universality results (of the second type) for orthogonal and symplectic ensembles.