  One of the most fundamental questions we can ask about a given gauge theory
is its phase diagram. In the standard model, we observe three fundamentally
different types of behavior: QCD is in a confined phase at zero temperature,
while the electroweak sector of the standard model combines Coulomb and Higgs
phases. Our current understanding of the phase structure of gauge theories owes
much to the modern theory of phase transitions and critical phenomena, but has
developed into a subject of extensive study. After reviewing some fundamental
concepts of phase transitions and finite-temperature gauge theories, we discuss
some recent work that broadly extends our knowledge of the mechanisms that
determine the phase structure of gauge theories. A new class of models with a
rich phase structure has been discovered, generalizing our understanding of the
confinement-deconfinement transition in finite-temperature gauge theories.
Models in this class have space-time topologies with one or more compact
directions. On R^3 x S^1, the addition of double-trace deformations or periodic
adjoint fermions to a gauge theory can yield a confined phase in the region
where the S^1 circumference L is small, so that the coupling constant is small,
and semiclassical methods are applicable. In this region, Euclidean monopole
solutions, which are constituents of finite-temperature instantons, play a
crucial role in the calculation of a non-perturbative string tension. We review
the techniques use to analyze this new class of models and the results obtained
so far, as well as their application to finite-temperature phase structure,
conformal phases of gauge theories and the large-N limit.
