  Active systems, or active matter, are self-driven systems which live, or
function, far from equilibrium - a paradigmatic example which we focus on here
is provided by a suspension of self-motile particles. Active systems are far
from equilibrium because their microscopic constituents constantly consume
energy from the environment in order to do work, for instance to propel
themselves. The nonequilibrium nature of active matter leads to a variety of
non-trivial intriguing phenomena. An important one which has recently been the
subject of intense interest among biological and soft matter physicists is that
of the so-called "motility-induced phase separation", whereby self-propelled
particles accumulate into clusters in the absence of any explicit attractive
interactions between them. Here we review the physics of motility-induced phase
separation, and discuss this phenomenon within the framework of the classic
physics of phase separation and coarsening. We also discuss theories for
bacterial colonies where coarsening may be arrested. Most of this work will
focus on the case of run-and-tumble and active Brownian particles in the
absence of solvent-mediated hydrodynamic interactions - we will briefly discuss
at the end their role, which is not currently fully understood in this context.
