L-functions and random matrices

In 1972 H. L. Montgomery announced a remarkable connection between the distribution of the zeros of the Riemann zeta-function and the distribution of eigenvalues of large random Hermitian matrices. Since then a number of startling developments have occurred making this connection more profound. In particular, random matrix theory has been found to be an extremely useful predictive tool in the theory of L-functions. In this article we will try to explain these recent developments and indicate some diretions for future investigations.