  Finite energy pluripotential theory accommodates the variational theory of
equations of complex Monge-Amp\`ere type arising in K\"ahler geometry. Recently
it has been discovered that many of the potential spaces involved have a rich
metric geometry, effectively turning the variational problems in question into
problems of infinite dimensional convex optimization, yielding existence
results for solutions of the underlying complex Monge-Amp\`ere equations. The
purpose of this survey is to describe these developments from basic principles.
