  I review some older work on the effective potentials of quantum field
theories, in particular the use of anomalous symmetries to constrain the form
of the effective potential, and the background field method for evaluating it
perturbatively. Similar techniques have recently been used to great success in
studying the effective superpotentials of supersymmetric gauge theories, and
one of my motivations is to present some of the older work on
non-supersymmetric theories to a new audience. The Gross-Neveu model exhibits
the essential features of the techniques. In particular, we see how rewriting
the Lagrangian in terms of an appropriate composite background field and
performing a perturbative loop expansion gives non-perturbative information
about the vacuum of the theory (the fermion condensate). The effective
potential for QED in a constant electromagnetic background field strength is
derived, and compared to the analogous calculation in non-Abelian Yang-Mills
theory. The Yang-Mills effective potential shows that the ``perturbative''
vacuum of Yang-Mills theory is unstable, and the true vacuum has a non-trivial
gauge field background. Finally, I describe how some of the limitations seen in
the non-supersymmetric theories are removed by supersymmetry, which allows for
exact computation of the effective superpotential in many cases.
