  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 in energy
(FAIR, RHIC-II and LHC). These are expected to provide key insights into
hadronic matter under extreme conditions using quarkonium observables.
