Torque magnetometry was used to investigate the magnetic anisotropy of epitaxial Fe3O4 films of thicknesses 420 and 40 nm grown by pulsed laser deposition on (0 0 1) MgO substrates. Torque measurements at 130 K in the (1 1̄ 0) plane allow us to evaluate the influence of shape and stress anisotropy. As in bulk material field-cooled measurements performed at T = 5 K show that the magnetic anisotropy is strongly influenced by the direction of the magnetic field applied above the Verwey temperature. In contrast to bulk magnetite we find smaller effective contributions of the magneto-crystalline anisotropy. In zero-field-cooled films our measurements suggest a tendency to a preferential out-of-plane orientation of the magnetic easy axis, but the dominating shape anisotropy favours an in-plane orientation of the magnetization.