Ultrathin L10 FePt films with different c-axis orientations relative to the film plane are promising candidates for data storage materials. In this work, within the framework of density functional theory, we calculated the magnetic properties of ultrathin L10 FePt (111) and (010) films with thicknesses ranging from 4 to 16 atomic monolayers (from about 0.8 to 3.5 nm). The highest average magnetic moments are observed for the thinnest films considered, and with increasing film thickness, the values converge towards the magnetic moment for bulk. The observed increase comes mainly from enhanced moments in the two atomic monolayers closest to the surface of the films. The easy axis of magnetization of (111) films prefers an alignment close to the tetragonal axis, an example of tilted magnetic anisotropy. The 6-monolayer ( 111) film (about 1.3 nm thick) inclines the easy axis of magnetization of about 45°to the film plane, which can find use in applications. The (010) films show an in-plane easy magnetization axis in a unique L10 tetragonal direction. This is an unusual type of in-plane anisotropy, as the particular direction preference is very strong. The computational results encourage further experimental studies of L10 systems with tilted and in-plane fixed magnetic anisotropy.