Diffractive proton-proton interactions, where one or two of the protons remain intact, are an important part of the physics programme at hadron colliders. On the one hand, they allow to check the diffractive formalism at very high energies, and on the other hand, they can serve as a search tool for new physics. In particular, the so-called central exclusive production process might provide a clean environment to search for new particles at the LHC.
The LHC magnets act as a precise longitudinal momentum spectrometer on the protons that have lost a fraction of their original momentum in diffractive scattering but remain intact. The forward proton detectors, (installed along the beam pipe at ~210 m from the central detector) are equipped with charged particle trackers to tag the intact protons and were brought online in 2017 and 2016 in ATLAS and CMS respectively. In this talk, I will discuss new research opportunities to become available with tagged forward protons using the LHC data.