We review the present status in the theoretical and phenomenological understanding of charmonium and bottomonium production in heavy-ion collisions. We start by recapitulating the basic notion of “anomalous quarkonium suppression” in heavy-ion collisions and its recent amendments involving regeneration reactions. We then survey in some detail concepts and ingredients needed for a comprehensive approach to utilize heavy quarkonia as a probe of hot and dense matter. The theoretical discussion encompasses recent lattice QCD computations of quarkonium properties in the Quark–Gluon Plasma, their interpretations using effective potential models, inelastic rate calculations and insights from analyses of electromagnetic plasmas. We illustrate the powerful techniques of thermodynamic Green functions ( T -matrices) to provide a general framework for implementing microscopic properties of heavy quarkonia into a kinetic theory of suppression and regeneration reactions. The theoretical concepts are tested in applications to heavy-ion reactions at SPS, RHIC and LHC. We outline perspectives for future experiments on charmonium and bottomonium production in heavy-ion collisions over a large range of energies (FAIR, RHIC-II and LHC). These are expected to provide key insights into hadronic matter under extreme conditions using quarkonium observables.