The structural, magnetic, transport, and magnetotransport properties of Fe3O4 thin films with thicknesses from 38 nm to 95 nm are systematically investigated. The occurrence of the Verwey transition in these films at a temperature of about 120 K is established. It is found that the temperature dependences of the magnetic moment have a feature near 40 K, which can be attributed to the multiferroic phase. According to the X-ray diffraction data, the film structure represents a (001) texture. As was established using transmission electron microscopy, the height and width of texture crystallites increase with film thickness. Analysis of the temperature dependences of the resistivity showed that the dominant carrier transport mechanism in the films is thermoactivated tunneling. The thermoactivation energy, along with the room-temperature resistivity, decreases with increasing film thicknesses, which is most likely related to the variation in the crystallite size, especially in the crystallite width. The field dependence of magnetoresistance behaves similarly over the entire temperature range and has a positive MR peak in weak fields, which is related to spin-dependent tunneling through Fe3O4 grains and antiferromagnetically coupled antiphase boundaries.