  A century ago, the foundations of equilibrium statistical mechanics were
laid. For a system in equilibrium with a thermal bath, much is understood
through the Boltzmann factor, exp{-H[C]/kT}, for the probability of finding the
system in any microscopic configuration C. In contrast, apart from some special
cases, little is known about the corresponding probabilities, if the same
system is in contact with more than one reservoir of energy, so that, even in
stationary states, there is a constant energy flux through our system. These
non-equilibrium steady states display many surprising properties. In
particular, even the simplest generalization of the Ising model offers a wealth
of unexpected phenomena. Mostly discovered through Monte Carlo simulations,
some of the novel properties are understood while many remain unexplained. A
brief review and some recent results will be presented, highlighting the sharp
contrasts between the equilibrium Ising system and this non-equilibrium
counterpart.
