The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. Particles may enter the system from the left with probability alpha, and exit from the right with probability beta. The model is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. In this talk, we will describe a surprising connection between the PASEP model and the combinatorics of certain 0-1 tableaux called permutation tableaux. Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration tau is a generating function for permutation tableaux of shape lambda(tau), enumerated according to three statistics. The tableaux in question come from total positivity on the Grassmannian (via work of Postnikov).