  Recent results on MSSM Higgs physics at the LHC are reviewed. The dependence
of the LHC discovery reach in the b\bar b H/A, H/A \to \tau^+\tau^- channel on
the underlying SUSY scenario is analysed. This is done by combining the latest
results for the prospective CMS experimental sensitivities for an integrated
luminosity of 30 or 60 fb^-1 with state-of-the-art theoretical predictions of
MSSM Higgs-boson properties. The results are interpreted in terms of the
parameters governing the MSSM Higgs sector at lowest order, M_A and tan_beta.
While the higgsino mass parameter \mu has a significant impact on the
prospective discovery reach (and correspondingly the ``LHC wedge'' region), it
is found that the discovery reach is rather stable with respect to variations
of other supersymmetric parameters. Within the discovery region a determination
of the masses of the heavy neutral Higgs bosons with an accuracy of 1-4% seems
feasible. It is furthermore shown that Higgs-boson production in central
exclusive diffractive channels can provide important information on the
properties of the neutral MSSM Higgs bosons.
