Charmonium production in Pb-Pb collisions at ALICE: from suppression to regeneration ?

<!--HTML-->Heavy quarkonium states are considered as one of the key observables for the study of the phase transition from a system made of hadrons towards a Quark-Gluon Plasma. In the last 25 years, experiments at CERN and Brookhaven have studied collisions of heavy ions looking for a suppression of charmonia/bottomonia, considered as a signature of the phase transition. After an introduction to the main concepts behind these studies, I will present the results obtained in Pb-Pb collisions by the ALICE experiment. The ALICE results will be critically compared to those of lower energy experiments, to CMS/ATLAS results, and to various model calculations. The large cross sections for heavy-quark production at LHC energies are expected to induce a novel production mechanism for charmonia in heavy-ion collisions, related to a (re)combination of ccbar pairs along the history of the collision and/or at hadronization. The occurrence of such a process at the LHC will be discussed. Finally, prospects for future measurements will be addressed.