Proton tagging at high luminosities at LHC

We show that forward proton taggers at the LHC installed at 220 and 420 m from ATLAS and / or CMS give access to a very rich forward physics program including possibilities to study the diffractive structure of proton or Higgs boson production in central exclusive diffraction processes, and also the photon-photon and photon-proton physics. The focus is put on projects aiming at tagging scattered protons at high luminosities at the LHC.