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.)