  In these lectures we present a brief review of the Higgs boson sector in the
``Standard Model'', and its Minimal Supersymmetric Extension, with particular
emphasis on the main mechanisms for Higgs production and decay at LEP2 and LHC,
and theoretical bounds on the Higgs boson masses. In the ``Standard Model'' the
effective potential can develop a non-standard minimum for values of the field
much larger than the weak scale. 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 which can be accomodated by the theory. In the ``Minimal
Supersymmetric Standard Model'', approximate analytical expressions for the
Higgs mass spectrum and couplings are worked out. An appropriate treatment of
squark decoupling allows to consider large values of the stop 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 LEP2 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.
