Abstract. The eigenvalue distributions of Gaussian random matrices and the random matrices from the classical groups play a fundamental role in the applications of random matrices. A basic question relates to the sampling from these distributions: how can it most efficiently be carried out? Rather than having to generate a random matrix of the sought type, and then computing its eigenvalues, it is now known that the characteristic polynomials in question satisfy simple recurrences with random coefficients. Thus the distributions can be sampled by computing the characteristic polynomials from the recurrences, and then computing its zeros. I'll review these developments, and explain my own contribution. One aspect of the latter (in joint work with Eric Rains) relates to the eigenvalue distribution of certain rank 1 perturbations, or equivalently the zeros of some random rational functions.