  We discuss different aspects of the Higgs self-interaction in the MS-bar and
the on-mass-shell (OMS) scheme. The running coupling \lambda(\mu) is
investigated in great detail. The three-loop coefficient of the \beta-function
in the OMS scheme is derived, and the three-loop running coupling is
calculated. The breakdown of perturbation theory for large Higgs masses M_H is
analyzed in three physical observables for which two-loop results are known.
Requiring the dependence on the renormalization scale to diminish
order-by-order in \lambda, we find that perturbation theory breaks down for
M_H=O(700 GeV) in Higgs decays. Similarly, M_H must be smaller than O(400 GeV)
for perturbatively calculated cross sections to be trustworthy up to cm
energies of O(2 TeV). If the Higgs sector shall be perturbative up to the GUT
scale, the Higgs must be lighter than O(150 GeV). For the two-loop observables
examined, the apparent convergence of the perturbation series is better in the
OMS scheme than in the MS-bar scheme.
