  Unlike equilibrium statistical mechanics, with its well-established
foundations, a similar widely-accepted framework for non-equilibrium
statistical mechanics (NESM) remains elusive. Here, we review some of the many
recent activities on NESM, focusing on some of the fundamental issues and
general aspects. Using the language of stochastic Markov processes, we
emphasize general properties of the evolution of configurational probabilities,
as described by master equations. Of particular interest are systems in which
the dynamics violate detailed balance, since such systems serve to model a wide
variety of phenomena in nature. We next review two distinct approaches for
investigating such problems. One approach focuses on models sufficiently simple
to allow us to find exact, analytic, non-trivial results. We provide detailed
mathematical analyses of a one-dimensional continuous-time lattice gas, the
totally asymmetric exclusion process (TASEP). It is regarded as a paradigmatic
model for NESM, much like the role the Ising model played for equilibrium
statistical mechanics. It is also the starting point for the second approach,
which attempts to include more realistic ingredients in order to be more
applicable to systems in nature. Restricting ourselves to the area of
biophysics and cellular biology, we review a number of models that are relevant
for transport phenomena. Successes and limitations of these simple models are
also highlighted.
