  We consider the general problem of determining the steady state of stochastic
nonequilibrium systems such as those that have been used to model (among other
things) biological transport and traffic flow. We begin with a broad overview
of this class of driven diffusive systems - which includes exclusion processes
- focusing on interesting physical properties, such as shocks and phase
transitions. We then turn our attention specifically to those models for which
the exact distribution of microstates in the steady state can be expressed in a
matrix product form. In addition to a gentle introduction to this matrix
product approach, how it works and how it relates to similar constructions that
arise in other physical contexts, we present a unified, pedagogical account of
the various means by which the statistical mechanical calculations of
macroscopic physical quantities are actually performed. We also review a number
of more advanced topics, including nonequilibrium free energy functionals, the
classification of exclusion processes involving multiple particle species,
existence proofs of a matrix product state for a given model and more
complicated variants of the matrix product state that allow various types of
parallel dynamics to be handled. We conclude with a brief discussion of open
problems for future research.
