The core-level binding energy of an atom in the surface region is different from that of the bulk atoms. From the measurement of this core-level shift many information on the electronic states and surface properties can be devised. The interpretation of the surface core-level shift in terms of the one-electron picture is not a simple task, because the core-level binding energy is correctly viewed as an energy difference between the ground state and a core-hole final state of the system. This total energy difference, that determines the core-level binding energy, is not simply related to the single-particle eigenvalues, due to relaxation and correlation effects. For this reason single-particle eigenvalues are only a gross approximation to core-level binding energy. On the contrary the core-level binding energy shift, on going from the surface to the bulk environment, can be successfully related to the single-particle eigenvalues shift. We will show this for various surface systems: (i) a clean surface of a metal, Ni; (ii) an interface between a metallic monolayer and a semiconductor, Na-Si(111); (iii) a semiconductor- insulator interface, F-Ca-Si(111). The investigation of these interfaces is based on the surface linear muttin-tin orbital method. Our results yield an accurate analysis of both valence and core electron states and of the nature of the interface bond.