Recent results on MSSM Higgs physics at the LHC are reviewed. The dependence of the LHC discovery reach in the bbar 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.
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