In about a third of proton-proton collisions in the Large Hadron Collider (LHC) at least one of the interacting protons survives and scatters to a very small angle. Detecting such forward protons extends the event characterisation and thus the LHC physics programme. Most often, the events with a surviving proton(s) are associated with low momentum transfer between the interacting protons (e.g. elastic or diffractive scattering). This corresponds to the nonperturbative regime of Quantum Chromodynamics (QCD), where first-principle theoretical predictions are difficult to obtain and largely missing. The experimental efforts thus consist in exciting exploratory work. Furthermore, the objects exchanged by the protons are expected to be gluon-rich and therefore provide a favourable search opportunity for "glueballs", composite particles built of gluons without valence quarks, as predicted e.g. by lattice QCD. Intact protons can also emerge from interactions with large momentum transfer. In such events one can probe centrally produced states with masses on the TeV scale - the events are thus interesting for searches of phenomena beyond the Standard Model of particle physics. The measurement of the forward protons can be a key ingredient. In exclusive processes, the kinematic correlation between the forward protons and the central system can yield a very strong background suppression and allow for precision measurements. Similarly, it simplifies the identification of possibly present "invisible" collision products, e.g. dark matter particles. At the LHC there are several experiments equipped with forward-proton detectors: ATLAS and CMS + TOTEM. By construction they have similar capabilities and in my talk I will illustrate the forward-physics potential with the last two. Besides others, I will show the recent experimental hints for the existence of “Odderon”, an object predicted in the 1970's but so far eluding a firm experimental confirmation.
A typical forward-physics event: two LHC protons (left) survive the collision and are detected with “Roman Pot” detectors (right top and bottom) while the system X, if produced, is detected with a multipurpose detector covering central rapidities (e.g. CMS, right middle).