[en] This paper discusses theoretical approaches for investigating Compton scattering from bound atomic electrons. It particularly emphasizes the understanding of methods currently used to obtain the scattered photon energy distribution at fixed scattering angle. This distribution is dominated by three features: the Compton line or peak, the resonant-Raman-Compton scattering and the infrared divergence at soft scattered photon energies. No single common approximation accesses all three regions, though they are all exhibited analytically in the nonrelativstic hydrogenic case. S-matrix calculations describe all of these features. The usual approximations utilized for Compton scattering are discussed and assessed in comparison with the more exact, relativistic S-matrix calculations. Finally, some recent developments in the theory of Compton scattering from bound electrons are noted. (author)