Pd/Co and Pd/Fe multilayer thin films containing ultrathin (2-12 Å) Co and Fe layers were prepared by vapour deposition in ultrahigh vacuum. Their magnetization was measured at room temperature in fields parallel and perpendicular to the film plane. The Pd/Co multilayers show a transition of the preferred direction from lying in the film plane towards lying along the film normal when decreasing the Co layer thickness below 8 Å. The Pd/Fe multilayers are preferably magnetized in the film plane, although the anisotropy decreases with lower Fe layer thickness. The magnetic anisotropy of both types of films can be interpreted by assuming an interface contribution which favours a perpendicular magnetization, and a volume contribution which favours an in-plane magnetization. The magnitude of the volume contribution is discussed in terms of magnetostatic, magnetocrystalline and magnetoelastic anisotropies, and equals, surprisingly, the anisotropy of the ferromagnetic elements in bulk thin film form in both cases.