We report on published charmonium measurements performed by ALICE, at the LHC, in Pb – Pb collisions at a center of mass energy per nucleon-nucleon collision sNN=2.76 TeV, at both mid (|y|<0.8) and forward (2.5<y<4) rapidities. The nuclear modification factor of inclusive J/ψ is presented as a function of the collision centrality and the J/ψ transverse momentum, pT. The variation of the J/ψ mean transverse momentum square as a function of the collision centrality is also discussed. These measurements are compared to state of the art models that include one or several of the following mechanisms: color screening of the charm quarks, statistical hadronization at the QGP phase boundary, balance between J/ψ dissociation and regeneration in the QGP, J/ψ interaction with a dense comoving medium. Results on the production of the heavier and less bound ψ(2S) meson in Pb–Pb collisions at forward-rapidity are also presented and compared to both models and measurements performed by other experiments. At mid-rapidity we also report on ALICE unique capability to separate prompt and non-prompt J/ψ production down to low pT(≥1.5 GeV/c) and thus disentangle between effects on prompt J/ψ mesons and energy loss of b quarks in the QGP.