  We outline the many quarkonium-physics opportunities offered by a
multi-purpose fixed-target experiment using the p and Pb LHC beams extracted by
a bent crystal. This provides an integrated luminosity of 0.5 fb-1 per year on
a typical 1cm-long target. Such an extraction mode does not alter the
performance of the collider experiments at the LHC. With such a high
luminosity, one can analyse quarkonium production in great details in pp, pd
and pA collisions at sqrt(sNN)~115 GeV and at sqrt(sNN)~72 GeV in PbA
collisions. In a typical pp (pA) run, the obtained quarkonium yields per unit
of rapidity are 2-3 orders of magnitude larger than those expected at RHIC and
about respectively 10 (70) times larger than for ALICE. In PbA, they are
comparable. By instrumenting the target-rapidity region, the large negative-xF
domain can be accessed for the first time, greatly extending previous
measurements by Hera-B and E866. Such analyses should help resolving the
quarkonium-production controversies and clear the way for gluon PDF extraction
via quarkonium studies. The nuclear target-species versatility provides a
unique opportunity to study nuclear matter and the features of the hot and
dense matter formed in PbA collisions. A polarised proton target allows the
study of transverse-spin asymmetries in J/psi and Upsilon production, providing
access to the gluon and charm Sivers functions.
