This paper reviews developments of the Brueckner theory of nuclei, starting with the multiple-scattering approach in the 1954 papers of Brueckner et al. Inevitably, the presentation represents the authors personal view on the problems. Theories rather than calculations have been emphasized. The latter depend very much upon the two-nucleon interaction which is still not known in sufficient detail. The definition of the Brueckner reaction-matrix is discussed at some length with regard to insertions in hole- and particle-lines. Löwdin's Exact Self-Consistent-Field theory that provides an exact many-body reaction matrix is given a short presentation for comparison with Brueckner's theory as applied to finite nuclei. Considerable attention has been given to the shell-model potential required for calculations on finite nuclei. The concept of single-particle energies is discussed in some detail in relation to removal energies. Some numerical results are reviewed either for illustrating the theory or because they are recent.