The asymmetric exclusion process (ASEP) is an important and well-studied non-equilibrium model from statistical physics, in which particles hop left and right on a finite one-dimensional lattice with open boundaries. Typically the model has 3 hopping parameters (governing the rates at which particles enter and exit the lattice, and hop left and right). The two-species ASEP is a generalization of the ASEP in which there are two types of particles, one "heavy" and one "light." Much past work was devoted to finding combinatorial formulas for the steady state probabilities of the original ASEP. In this talk I will present our recent work on the combinatorics of the two-species ASEP. Together with Viennot, we introduced "rhombic alternative tableaux," which are fillings of certain rhombic tilings, and used them to provide combinatorial formulas for the two-species ASEP. Furthermore, in recent joint work with Corteel and Williams, we introduced "rhombic staircase tableaux," and provided combinatorial formulas for the more general 5-parameter two-species ASEP, which has an interesting connection to Koornwinder-Macdonald polynomials. The talk will be accessible to graduate students.