How can a heavy Higgs boson be consistent with the precision electroweak measurements?

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.