The author reviews some of the important successes achieved by Eliashberg theory in describing the observed superconducting properties of many conventional superconductors. Functional derivative techniques are found to help greatly in understanding the observed deviations from BCS laws. Approximate analytic formulas with simple correction factors for strong-coupling corrections embodied in the single parameter $\frac{T_c}{{\omega }_{\ln }}$ are also found to be very helpful. Here $T_c$ is the critical temperature and ${\omega }_{\ln }$ is an average boson energy mediating the pairing potential in Eliashberg theory. In view of the discovery of high-$T_c$ superconductivity in the copper oxides, results in the very strong coupling limit of $\frac{T_c}{{\omega }_{\ln }}\sim 1$ are also considered, as is the asymptotic limit when $\frac{T_c}{{\omega }_{\ln }}\to \infty$. This case is of theoretical interest only, but it is nevertheless important because simple analytic results apply that give insight into the more realistic strong-coupling regime. A discussion more specific to the oxides is included in which it is concluded that some high-energy boson-exchange mechanism must be operative, with, possibly, some important phonon contribution in some cases. A more definitive application of boson-exchange models to the oxides awaits better experimental results.

DOI:https://doi.org/10.1103/RevModPhys.62.1027