Prompt $J/\psi$ production at the LHC

Models with essential non-perturbative QCD dynamics for the production of charmonium are extrapolated to give predictions of prompt J/psi production at the LHC. Differences of up to an order of magnitude occurs, although the different models all describe the Tevatron data on high-pT charmonium. An important point is here the treatment of higher order perturbative QCD effects. We also discuss the large rate of prompt J/psi from these models as a background to CP violation studies.