Diffractive processes as a tool for searching for new physics

We show that the addition of detectors to tag the outgoing forward protons, at the LHC, will significantly enlarge the potential of studying New Physics. A topical example is Higgs production by the exclusive double-diffractive process, pp -> p+H+p. We discuss the production of Higgs bosons in both the SM and MSSM. We show how the predicted rates may be checked at the Tevatron by observing the exclusive double-diffractive production of dijets, or $\chi_c$ or $\chi_b$ mesons, or $\gamma \gamma$ pairs.