Photoemission from a many-electron system is a many-electron process, even though the transition operator may affect only one electron directly. Relaxation and “shake-up” structure are related by a sum rule: when one is present, the other must also be present. Shake-up structure is shown to be accurately predictable in atomic neon and molecular HF if the CI calculations are done carefully. In metals the sum rule also applies but final-state effects usually appear as relaxation energy, which is large even for valence electrons. Finally, in rare-earth metals discrete shake-up structure is observable in the 4p region.