Ferromagnetic materials exhibit intrinsic `easy' and `hard' directions of the magnetization. This magnetic anisotropy is, from both a technological and fundamental viewpoint one of the most important properties of magnetic materials. The magnetic anisotropy in metallic magnetic multilayers forms the subject of this review article. As individual layers in a multilayer stack become thinner, the role of interfaces and surfaces may dominate that of the bulk: this is the case in many magnetic multilayers, where a perpendicular interface contribution to the magnetic anisotropy is capable of rotating the easy magnetization direction from in the film plane to perpendicular to the film plane. In this review, we show that the (in-plane) volume and (perpendicular) interface contribution to the magnetic anisotropy have been separated into terms related to mechanical stresses, crystallographic structure and the planar shape of the films. In addition, the effect of roughness, often inherent to the deposition techniques used, has been addressed theoretically. Several techniques to prepare multilayers and to characterize their growth as well as methods to determine the magnetic anisotropy are discussed. A comprehensive survey of experimental studies on the perpendicular magnetic anisotropy in metallic multilayers containing Fe, Co or Ni is presented and commented on. Two major subjects of this review are the extrinsic effects of strain, roughness and interdiffusion and the intrinsic effect of the crystallographic orientation on the magnetic anisotropy. Both effects are investigated with the help of some dedicated experimental studies. The results of the orientational dependence studies are compared with ab initio calculations. Finally, the perpendicular surface anisotropy and the in-plane step anisotropy are discussed.