The Sachdev-Ye-Kitaev model, which consists of a large number of fermions with all-to-all random interactions, is a solvable toy model of quantum chaos. We present an analysis of supersymmetric generalizations of this model using tools borrowed from random matrix theory and free probability. This allows us to compute the asymptotic density of states of these models, as well as 2-point correlation functions at all energy scales. We relate these results to q-brownian motion, quantum groups, and 2-d supergravity.