We introduce some new tableaux, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. In its most general form, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all five parameters general, in terms of tableaux. Our second result is a related combinatorial formula for the moments of Askey-Wilson polynomials. Since the 1980's there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials. However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials. This is joint work with Sylvie Corteel.