  The upper limit on the lightest CP-even Higgs boson mass, mh, is analyzed
within the MSSM as a function of tan(beta) for fixed mtop and Msusy. The impact
of recent diagrammatic two-loop results on this limit is investigated. We
compare the MSSM theoretical upper bound on mh with the lower bound obtained
from experimental searches at LEP. We estimate that with the LEP data taken
until the end of 1999, the region mh < 108.2 GeV can be excluded at the 95%
confidence level. This corresponds to an excluded region 0.6 <= tan(beta) <=
1.9 within the MSSM for mtop = 174.3 GeV and Msusy <= 1 TeV. The final
exclusion sensitivity after the end of LEP, in the year 2000, is also briefly
discussed. Finally, we determine the upper limit on mh within the Minimal
Supergravity (M-SUGRA) scenario up to the two-loop level, consistent with
radiative electroweak symmetry breaking. We find an upper bound of mh \approx
127 GeV for mtop = 174.3 GeV in this scenario, which is slightly below the
bound in the unconstrained MSSM.
