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

Quarkonium Physics at a Fixed-Target Experiment using the LHC Beams