the beta ensembles of random matrix theory are natural generalizations of the gaussian orthogonal, unitary, and symplectic ensembles, these classical cases corresponding to beta = 1, 2, or 4. we prove that the largest eigenvalues in the general ensembles have limit laws described by the low lying spectrum of certain random schroedinger operators, providing a new characterization of the celebrated tracy-widom laws. as a corollary, a second characterization is available via the explosion probability of an associated one-dimensional diffusion. a complementary picture is developed for beta versions of random sample-covariance matrices. (based on work with j. ramirez and b. virag.)