  The fit of precision electroweak data to the Minimal Standard Model currently
gives an upper limit on the Higgs boson mass of 170 GeV at 95% confidence.
Nevertheless, it is often said that the Higgs boson could be much heavier in
more general models. In this paper, we critically review models that have been
proposed in the literature that allow a heavy Higgs boson consistent with the
precision electroweak constraints. All have unusual features, and all can be
distinguished from the Minimal Standard Model either by improved precision
measurements or by other signatures accessible to next-generation colliders.
