  The status of the triviality bound of the Higgs mass in the Minimal Standard
Model is reviewed. It is emphasized that the bound is obtained, in the scalar
sector, by limiting cutoff effects on physical processes. Results from several
regularization schemes, including actions that allow a parameterization and
tuning of the leading cutoff effects, are presented. They lead to the
conclusion that the Minimal Standard Model will describe physics to an accuracy
of a few percent up to energies of the order 2 to 4 times the Higgs mass,
$M_H$, only if $M_H \le 710 \pm 60 ~ GeV$. The status of Higgs and fermion mass
bounds in Higgs-fermion models is also briefly reviewed.
