The exclusion process is a well studied interacting particle system
that can be regarded as a model for the stochastic evolution of infinite systems
from various contexts, including statistical physics, traffic flow, and
the study of biopolymers. In this talk, I will survey the current state of
our understanding of the structure of the set of stationary distributions of
these processes. For symmetric systems, we have had a complete characterzation
for over thirty years. Asymmetric systems are much more challenging. I will
present fairly complete results in one dimension, will describe the much
less satisfactory picture in higher dimensions, and will discuss several
open problems and conjectures. Much of this work is joint with Maury Bramson,
Lincoln Chayes, and Tom Mountford
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