The experimental proof that the superconductivity in metals such as P b, Sn, Nb , etc. is induced by electron-phonon interaction rests on analysis of tunneling spectrum [12, 13] in these metals using the Eliashberg theory and the measurement of the spectrum of phonons by neutron scattering. The theory also provides experimental proofs of its limit of validity. With one important modification, which does not affect the linearized Eliashberg theory which is enough to determine T$_{c}$ , the theory can be used for pairing in any symmetry of degenerate fermions due to exchange of any kind of fluctuations, provided they can be regarded as bosons, and provided the Migdal condition is satisfied, where ω$_{c}$ is the upper cutoff of the spectrum of bosons. Excellent reviews of the technical aspects of the Eliashberg theory have been written [14]. To fulfill the limited goals of this review, I need to present only the dependence of some of the results of the theory on the parameters of the starting model. The model specifies a spectrum of fermions, a spectrum of bosons, and the interactions between the fermions and the bosons: The transition temperature is the temperature at which the particle-particle scattering vertex with tot al momentum 0 and energy 0 diverges. The exact v ertex Γ$_{S}$( k, k + q ) for fermions with tota l spin S = 0 , 1, scattering from k , − k to k + q , − k − q , with energy ω to ω + ν follo ws the Bethe-Salpeter equation through which it is related to the irreducible particle-particle vertex I$_{S}$ ( k, k + q ), as shown in the Fig.(2): Γ$_{S}$( k, k + q ) = I$_{S}$ ( k, k + q ) − ( T/ 2) ∑ p ,ω I$_{S}$ ( k, p ) G ( p , ω ) G ( − p , − ω )Γ$_{S}$ ( p, k + q ) . (2) H$_{f}$ = ∑ k ,σ ϵ$_{k}$c + k $_{σ}$c$_{k}$$_{σ}$ (1) FIG. 2: Bethe-Salp eter Equation for the relation between the complete particle-particle vertex Γ and the irred ucible vertex I . = + Γ Γ k k k k ′ k ′ k ′ k ′ k ′ k ′ k − k − k I p − p I S S S S 12