We review the present status in the theoretical and phenomenologicalunderstanding of charmonium and bottomonium production in heavy-ion collisions.We start by recapitulating the basic notion of "anomalous quarkoniumsuppression" in heavy-ion collisions and its recent amendments involvingregeneration reactions. We then survey in some detail concepts and ingredientsneeded for a comprehensive approach to utilize heavy quarkonia as a probe ofhot and dense matter. The theoretical discussion encompasses recent lattice QCDcomputations of quarkonium properties in the Quark-Gluon Plasma, theirinterpretations using effective potential models, inelastic rate calculationsand insights from analyses of electromagnetic plasmas. We illustrate thepowerful techniques of thermodynamic Green functions (T-matrices) to provide ageneral framework for implementing microscopic properties of heavy quarkoniainto a kinetic theory of suppression and regeneration reactions. Thetheoretical concepts are tested in applications to heavy-ion reactions at SPS,RHIC and LHC. We outline perspectives for future experiments on charmonium andbottomonium production in heavy-ion collisions over a large range in energy(FAIR, RHIC-II and LHC). These are expected to provide key insights intohadronic matter under extreme conditions using quarkonium observables.
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