Many-body theory of electron correlations in atoms: RPAE and beyond

It is demonstrated how the correlations of electrons manifest themselves in photoionization of atoms. The diagrammatical technique, convenient and transparent, is applied to study this and related processes. Choosing as the best one particle the Hartree-Fock approximation, the first considerable step in accounting for electron correlations is made by constructing the Random Phase Approximation with Exchange. Its generalizations are also described, which include rearrangement of electron shells due to vacancies creation and decay. Attention is given to ''two electron-two vacancy'' excitations, formation of the negative ions and their photoionization as well as to satellites and ''shadows''. The direct knock-out of secondary particles, electrons and photons, by photoelectrons is considered. Formation of multiply-charged ions and above threshold phenomena, mainly multistep PCI, are discussed. Future of the domain: new atom-like objects and next steps in theoretical studies are outlined. (author)