  We report on the spin and diffractive physics at a future multi-purpose
fixed-target experiment with proton and lead 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 using typical targets would
surpass that of RHIC by more than 3 orders of magnitude. The fixed-target mode
has the advantage to allow for measurements of single-spin asymmetries with
polarized target as well as of single-diffractive processes in the target
region.
