Prompt/Non-prompt <i>J/ψ</i> production in proton-proton and Pb-Pb collisions with ALICE

Quarkonium production in high-energy hadronic collisions is sensitive to both perturbative and non-perturbative aspects of QCD calculations. The charmonium production cross section can be split into prompt and non-prompt components, the first corresponding to a direct production of (anti-)charm quarks, the second originating from the decay of beauty hadrons. The latter is important to investigate the mass dependence of heavy-quarks in-medium energy-loss mechanism. In this contribution the recent measurements of prompt and non-prompt J/ψ carried out by the ALICE Collaboration in pp and Pb–Pb collisions at midrapidity (|y| &lt; 0.8) will be presented. Moreover, thanks to the installation of the new muon forward tracker (MFT), the prompt/non-prompt charmonia separation is possible in LHC Run 3 also at forward rapidity (2.5 &lt; y &lt; 4). The status of the new measurements will be presented.