  We report on the opportunities for spin physics and Transverse-Momentum
Dependent distribution (TMD) studies at a future multi-purpose fixed-target
experiment using the proton or lead ion LHC beams extracted by a bent crystal.
The LHC multi-TeV beams allow for the most energetic fixed-target experiments
ever performed, opening new domains of particle and nuclear physics and
complementing that of collider physics, in particular that of RHIC and the EIC
projects. The luminosity achievable with AFTER@LHC using typical targets would
surpass that of RHIC by more that 3 orders of magnitude in a similar energy
region. In unpolarised proton-proton collisions, AFTER@LHC allows for
measurements of TMDs such as the Boer-Mulders quark distributions, the
distribution of unpolarised and linearly polarised gluons in unpolarised
protons. Using the polarisation of hydrogen and nuclear targets, one can
measure transverse single-spin asymmetries of quark and gluon sensitive probes,
such as, respectively, Drell-Yan pair and quarkonium production. The
fixed-target mode has the advantage to allow for measurements in the
target-rapidity region, namely at large x^uparrow in the polarised nucleon.
Overall, this allows for an ambitious spin program which we outline here.

High Energy Physics - Experiment;High Energy Physics - Phenomenology;Nuclear Experiment;Nuclear Theory