We report on a future multi-purpose fixed-target experiment with the proton or lead ion LHC beams extracted by a bent crystal. The multi-TeV LHC beams allow for the most energetic fixed-target experiments ever performed. Such an experiment, tentatively named AFTER for "A Fixed-Target ExperRiment", gives access to new domains of particle and nuclear physics complementing that of collider experiments, in particular at RHIC and at the EIC projects. The instantaneous luminosity at AFTER using typical targets surpasses that of RHIC by more than 3 orders of magnitude. Beam extraction by a bent crystal offers an ideal way to obtain a clean and very collimated high-energy beam, without decreasing the performance of the LHC. The fixed-target mode also has the advantage of allowing for spin measurements with a polarised target and for an access over the full backward rapidity domain up to xF ~ - 1. Here, we elaborate on the reachable luminosities, the target polarisation, and a selection of measurements with hydrogen and deuterium targets. Show Partial Abstract