Prospects for quarkonium-production studies accessible during the upcoming high-luminosity phases of the CERN Large Hadron Collider operation after 2021 are reviewed. Current experimental and theoretical open issues in the field are assessed together with the potential for future studies in quarkonium-related physics. This will be possible through the exploitation of the huge data samples to be collected in proton–proton, proton–nucleus and nucleus–nucleus collisions, both in the collider and fixed-target modes. Such investigations include, among others, those of: (i) J/ψ and Υ produced in association with other hard particles; (ii) χc,b and ηc,b down to small transverse momenta; (iii) the constraints brought in by quarkonia on gluon PDFs, nuclear PDFs, TMDs, GPDs and GTMDs, as well as on the low-x parton dynamics; (iv) the gluon Sivers effect in polarised-nucleon collisions; (v) the properties of the quark–gluon plasma produced in ultra-relativistic heavy-ion collisions and of collective partonic effects in general; and (vi) double and triple parton scatterings.