  There has been significant progress recently in our understanding of the
stationary measures of the exclusion process on $Z$. The corresponding
situation in higher dimensions remains largely a mystery. In this paper we give
necessary and sufficient conditions for a product measure to be stationary for
the exclusion process on an arbitrary set, and apply this result to find
examples on $Z^d$ and on homogeneous trees in which product measures are
stationary even when they are neither homogeneous nor reversible. We then begin
the task of narrowing down the possibilities for existence of other stationary
measures for the process on $Z^d$. In particular, we study stationary measures
that are invariant under translations in all directions orthogonal to a fixed
nonzero vector. We then prove a number of convergence results as $t\to\infty$
for the measure of the exclusion process. Under appropriate initial conditions,
we show convergence of such measures to the above stationary measures. We also
employ hydrodynamics to provide further examples of convergence.
