Many complex systems of interacting particles behave on a phenomenological level in some random fashion. Examples come from areas as diverse as the growth of surfaces or the growth of biological systems, reaction-diffusion processes or the study of traffic flow. However, general theory for systems far from thermal equilibrium is still being developed. Hence one usually investigates specific model systems, hoping to gain insight into the general behavior of such systems.
However, there is an important class of so called asymmetric simple exclusion processes (ASEP) which allow analytic treatment. It is commonly accepted that different versions of ASEP provide an adequate description of statistical properties of one-dimensional diffusive and driven-diffusive systems. During the last decade the ASEP was a laboratory for obtaining the universal critical exponents and scaling functions for different universality classes. Surprisingly recent results from integrable systems and representation theory of quantum groups found applications in stochastic processes where they define the most general ASEP process with exactly solvable dynamics.