  Some important rigorous results on phase transitions accompanied by the
spontaneous breaking of symmetries in statistical mechanics and relativistic
quantum field theory are reviewed. Basic ideas, mainly inspired by quantum
field theory, underlying the proofs of some of these results are sketched. The
Goldstone theorem is proven, and the Mermin-Wagner-Hohenberg theorem concerning
the absence of continuous symmetry breaking in one and two dimensions is
recalled. Comments concerning rigorous results on the Kosterlitz-Thouless
transition in the two-dimensional classical XY model are made.
