A review of the theory of electron momentum spectroscopy (EMS) based on the single (multiple) electron impact ionization of a quantum target (atom, molecule, thin film) is presented. Numerous examples are considered which demonstrate the advantages of EMS for interpreting the momentum distributions of the ejected electrons compared to other methods used for direct studies of the many-body wave function of a target. In particular, it is shown that EMS provides considerable progress in the quantum chemistry of molecules and gives a better understanding of processes leading to rearrangement of their outer electron orbitals upon their adiabatic approach. In addition, EMS allows one to study in detail the band structure and form of one-electron wave functions for both crystalline and amorphous thin films. The possibility of direct ’portraying’ of electron correlations in atoms using the double ionization technique is also considered.