By extracting the beam with a bent crystal or by using an internal gas target, the multi-TeV proton and lead LHC beams allow one to perform the most energetic fixed-target experiments (AFTER@LHC) and to study $p$+$p$ and $p$+A collisions at $\sqrt{s_{NN}}$=115 GeV and Pb+$p$ and Pb+A collisions at $\sqrt{s_{NN}}$=72 GeV. Such studies would address open questions in the domain of the nucleon and nucleus partonic structure at high-$x$, quark-gluon plasma and, by using longitudinally or transversally polarised targets, spin physics.
In this paper, we discuss the physics opportunities of a fixed-target experiment at the LHC and we report on the possible technical implementations of a high-luminosity experiment. We finally present feasibility studies for Drell-Yan, open heavy-flavour and quarkonium production, with an emphasis on high-$x$ and spin physics.