The Higgs boson search has shifted from LEP2 to the Tevatron and will subsequently move to the LHC. The current limits from the Tevatron and the prospective sensitivities at the LHC are often interpreted in specific MSSM scenarios. For heavy Higgs boson production and subsequent decay into b{bar b} or??−, the present Tevatron data allow to set limits in the M{sub A}-tan? plane for small M{sub A} and large tan? values. Similar channels have been explored for the LHC, where the discovery reach extends to higher values of M{sub A} and smaller tan?. Searches for MSSM charged Higgs bosons, produced in top decays or in association with top quarks, have also been investigated at the Tevatron and the LHC. We analyze the current Tevatron limits and prospective LHC sensitivities. We discuss how robust they are with respect to variations of the other MSSM parameters and possible improvements of the theoretical predictions for Higgs boson production and decay. It is shown that the inclusion of supersymmetric radiative corrections to the production cross sections and decay widths leads to important modifications of the present limits on the MSSM parameter space. The impact on the region where only the lightest MSSM Higgs boson can be detected at the LHC is also analyzed. We propose to extend the existing benchmark scenarios by including additional values of the higgsino mass parameter?. This affects only slightly the search channels for a SM-like Higgs boson, while having a major impact on the searches for non-standard MSSM Higgs bosons.