  Just over fifty years ago, Pierre Hohenberg developed a rigorous proof of the
non-existence of long-range order in a two-dimensional superfluid or
superconductor at finite temperatures. The proof was immediately extended by N.
D. Mermin and H. Wagner to the Heisenberg ferromagnet and antiferromagnet, and
shortly thereafter, by Mermin to prove the absence of translational long-range
order in a two-dimensional crystal, whether in quantum or classical mechanics.
In this paper, we present an extension of the Hohenberg-Mermin-Wagner theorem
to give a rigorous proof of the impossibility of long-range ferromagnetic order
in an itinerant electron system without spin-orbit coupling or magnetic dipole
interactions. We also comment on some situations where there are compelling
arguments that long-range order is impossible but no rigorous proof has been
given, as well as situations, such as a magnet with long range interactions, or
orientational order in a two-dimensional crystal, where long-range order can
occur that breaks a continuous symmetry.
