Quarkonium production in high-energy reactions is found to exhibit a behaviour more universal than that expected from velocity scaling. Total rates of quarkonia produced in hadronic interactions as well as Feynmanx and transverse momentum distributions can be described over the full range of accessible energies (15GeV) by two-stage processes. The quarkonium production cross section factors into a process-dependent short-distance part and a single long-distance matrix element. The first part describing the production of a free quark-antiquark pair is the perturbatively calculated subthreshold cross section. The non-perturbative factor turns out to be universal, giving the model great predictive power. Furthermore we estimate the fraction of the heavy-quark cross section leading to quarkonium for both the charm and bottom systems. Finally, we comment on quarkonium photoproduction.a Heisenberg Fellow.