First-principles calculations of the magnetic anisotropy energies of Pd/Co metallic multilayers have been performed to investigate the effect of strain on the interface magnetic anisotropy. Also, to clarify the contribution of the interface to anisotropy, the anisotropy energies of an unsupported Co monolayer and bulk Co have been calculated. These two systems have different interfaces compared to the Pd/Co multilayer in the sense that an unsupported Co monolayer and bulk Co can be considered as vacuum/Co and Co/Co multilayers, respectively. A Pd/Co multilayer is predicted to exhibit a perpendicular magnetic anisotropy in accordance with experiment. Bulk Co shows perpendicular anisotropy, but the anisotropy energy is quite small compared to that of Pd/Co. On the contrary, an unsupported Co monolayer shows an in-plane anisotropy. These differences suggest the importance of the existence of the interface for perpendicular magnetic anisotropy, which originates from the modification of the local electronic structure of a Co layer due to the presence of the interface. The strength of the hybridization of electronic states at the interface determines the relative position of the Fermi energy to the position of the local density of states (LDOS) of |m|=2 character of Co d electrons of minority spin. If the LDOS of |m|=2 character is large at the Fermi energy, the system shows a perpendicular anisotropy. As for the effect of strain, the anisotropy energy of Pd/Co increases as a function of interatomic distance in the in-plane direction, while that of a Co monolayer decreases. Compared to these two systems, the magnetoelastic constant of bulk Co is considerably smaller. These results suggest that the effect of strain on the anisotropy energy is strongly correlated with the type of atomic species adjacent to the Co layer and cannot be determined solely from the value of strain introduced in the Co layer. In the case of a heterointerface, the strength of hybridization between the orbitals inside the monolayer and that between the orbitals in adjacent monolayers are quite different and this fact leads to the large strain dependence of the anisotropy. \textcopyright{} 1996 The American Physical Society.