We give an asymptotic evaluation of the complexity of spherical p-spin spin glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the ground state), and the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our approach allows us to compute the related TAPcomplexity and extend the results known in the physics literature. As an independent tool, we prove a large deviation principle for the kth-largest eigenvalue of the Gaussian orthogonal ensemble, extending the results of Ben Arous, Dembo, and Guionnet. © 2012 Wiley Periodicals, Inc.