The ALICE experiment has studied J/psi production at forward rapidity from pT=0, through its dimuon decay channel, in proton-proton and Pb-Pb collisions at the LHC. Results obtained in proton-proton collisions at sqrt(s)=7 TeV will be discussed. The integrated and differential (in pT and rapidity) inclusive production cross sections will be presented and compared to results from other LHC experiments. First results on the J/psi integrated and differential cross sections in proton-proton collisions at sqrt(s)=2.76 TeV will also be discussed. These measurements, at the same centre-of-mass energy as for Pb-Pb collisions, provide a crucial reference for the study of nuclear matter effects on J/psi production.