  We investigate charmonium production in the hot medium created by heavy-ion
collisions by setting up a framework in which in-medium charmonium properties
are constrained by thermal lattice QCD (lQCD) and subsequently implemented into
kinetic approaches. A Boltzmann transport equation is employed to describe the
time evolution of the charmonium phase space distribution with the loss and
gain term accounting for charmonium dissociation and regeneration (from charm
quarks), respectively. The momentum dependence of the charmonium dissociation
rate is worked out. The dominant process for in-medium charmonium regeneration
is found to be a 3-to-2 process. Its corresponding regeneration rates from
different input charm-quark momentum spectra are evaluated. Experimental data
on $J/\psi$ production at CERN-SPS and BNL-RHIC are compared with our numerical
results in terms of both rapidity-dependent inclusive yields and transverse
momentum ($p_t$) spectra. Within current uncertainties from (interpreting) lQCD
data and from input charm-quark spectra the centrality dependence of $J/\psi$
production at SPS and RHIC (for both mid- and forward rapidity) is reasonably
well reproduced. The $J/\psi$ $p_t$ data are shown to have a discriminating
power for in-medium charmonium properties as inferred from different
interpretations of lQCD results.
