These twenty-six expository papers on random matrices and products of random matrices survey the major results of the last thirty years. They reflect both theoretical and applied concerns in fields as diverse as computer science, probability theory, mathematical physics, and population biology. Many of the articles are tutorial, consisting of examples, sketches of proofs, and interpretations of results. They address a wide audience of mathematicians and scientists who have an elementary knowledge of probability theory and linear algebra, but not necessarily any prior exposure to this specialized area. More advanced articles, aimed at specialists in allied areas, survey current research with references to the original literature. The book's major topics include the computation and behavior under perturbation of Lyapunov exponents and the spectral theory of large random matrices.The applications to mathematical and physical sciences under consideration include computer image generation, card shuffling, and other random walks on groups, Markov chains in random environments, the random Schroedinger equations and random waves in random media. Most of the papers were originally presented at an AMS-IMS-SIAM Joint Summer Research Conference held at Bowdoin College in June, 1984. Of special note are the papers by Kotani on random Schroedinger equations, Yin and Bai on spectra for large random matrices, and Newman on the relations between the Lyapunov and eigenvalue spectra.