  After reviewing briefly the upper bounds on the mass of the lightest Higgs
boson in the most general unconstrained minimal supersymmetric extension of the
Standard Model, we discuss various arguments which reduce the parameter space
of the model and give stronger predictions for $M_{h^0}$. First, the
constraints from the presently available experimental data are summarized.
Next, the role of of several additional theoretical assumptions is studied,
after extrapolating the model to high energy scales. The most important ones
are: perturbative validity up to the GUT scale and the electroweak symmetry
breaking. A Higgs boson with $M_{h^0}<$100 GeV is predicted in several
scenarios. Its absence in that mass range will have important implications for
the parameter space of the model.
