The asymmetric simple exclusion process (ASEP) is a continuous time Markov process of interacting particles on a lattice . ASEP is dened by two rules: (1) A particle at x waits an exponential time with parameter one, and then chooses y with probability p(x,y); (2) If y is vacant at that time it moves to y, while if y is occupied it remains at x. The main interest lies in innite particle systems. In this lecture we consider the ASEP on the integer lattice Z with nearest neighbor jump rule: p(x, x + 1) = p, p(x, x-1) = 1-p and p œ. The integrable structure is that of Bethe Ansatz. We discuss various limit theorems which in certain cases establishes KPZ universality.