The asymmetric simple exclusion process (ASEP) is a model from statistical physics that describes the dynamics of particles hopping right and left on a finite 1-dimensional lattice. Particles can enter and exit at the left and right boundaries, and at most one particle can occupy each site. The ASEP plays an important role in the study of non-equilibrium statistical mechanics and has appeared in many contexts, for instance as a model for 1-dimensional transport processes such as protein synthesis, molecular and cellular transport, and traffic flow. Moreover, it displays rich combinatorial structure: one can compute the stationary probabilities for the ASEP using fillings of certain tableaux. In this talk, we will discuss some of the combinatorial results from the past decade as well as recent developments, including combinatorial formulae for a two-species generalization of the ASEP and a remarkable connection to orthogonal polynomials. This talk is based on joint works with X. Viennot and separately with S. Corteel and L. Williams.