The inelastic production of J/ψ mesons in ep collisions has been studied with the ZEUS detector at HERA using an integrated luminosity of 109 pb-1. The J/ψ mesons were identified using the decay channel J/ψ → μ+μ- The measurements were performed in the kinematic range 2 < Q2 < 80 GeV2, 50 < W < 250 GeV, 0.2 < z < 0.9 and -1.6 < Ylab < 1.3, where Q2 is the virtuality of the exchanged photon, W is the photon-proton centre-of-mass energy, z is the fraction of the photon energy carried by the J/ψ meson in the proton rest frame and Ylab is the rapidity of the J/ψ in the laboratory frame. The measured cross sections are compared to theoretical predictions within the non-relativistic QCD framework including colour-singlet and colour-octet contributions, as well as to predictions based on the κT-factorisation approach. Calculations of the colour-singlet process generally agree with the data, whereas inclusion of colour-octet terms spoils this agreement. © Springer-Verlag/Società Italiana di Fisica 2005.