  Depending on the Higgs-boson and top-quark masses, $M_H$ and $M_t$, the
effective potential of the {\bf Standard Model} can develop a non-standard
minimum for values of the field much larger than the weak scale. In those cases
the standard minimum becomes metastable and the possibility of decay to the
non-standard one arises. Comparison of the decay rate to the non-standard
minimum at finite (and zero) temperature with the corresponding expansion rate
of the Universe allows to identify the region, in the ($M_H$, $M_t$) plane,
where the Higgs field is sitting at the standard electroweak minimum. In the
{\bf Minimal Supersymmetric Standard Model}, approximate analytical expressions
for the Higgs mass spectrum and couplings are worked out, providing an
excellent approximation to the numerical results which include all
next-to-leading-log corrections. An appropriate treatment of squark decoupling
allows to consider large values of the stop and/or sbottom mixing parameters
and thus fix a reliable upper bound on the mass of the lightest CP-even Higgs
boson mass. The discovery of the Higgs boson at LEP~2 might put an upper bound
(below the Planck scale) on the scale of new physics $\Lambda$ and eventually
disentangle between the Standard Model and the Minimal Supersymmetric Standard
Model.
