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 ever and to study $pp$, $p$d and $p$A collisions at $\sqrt{s_{NN}}=115$ GeV and Pb$p$ and PbA collisions at $\sqrt{s_{NN}}=72$ GeV with high precision and modern detection techniques. 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 will review the technical solutions to obtain a high-luminosity fixed-target experiment at the LHC and will discuss their possible implementations with the ALICE and LHCb detectors.