A review is given of the nonrelativistic many-body perturbation theory (MBPT), and an analysis is made about the possibilities of extending this scheme to the relativistic regime in a rigorous and systematic manner. The nonrelativistic MBPT is here based upon the iterative solution of coupled one- and two-particle equations, which yields the polarization (one-particle) and pair-correlation (two-particle) effects to all orders of perturbation theory. Illustrative numerical results are given. The analysis of the relativistic problem is based upon the covariant treatment of the electron-electron interaction (Feynman gauge) with the exchange of one and two virtual photons. The first approximation leads to the so-called no-virtual-pair approximation, where the effect of virtual electron-positron pairs as well as Lamb-shift corrections are neglected. Here, a numerical procedure quite similar to that employed in the nonrelativistic case has been developed by the Chalmers group, and some results are given. The effects beyond the no-virtual-pair approximation cannot be taken into account in a complete and rigorous way by means of a single potential. For the pair-correlation problem at least two potentials are needed, and in addition there are three-body potentials etc. Furthermore, these potentials are energy dependent, so an exact calculation seems to require explicit construction of the intermediate states.