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We analyze the validity of Eliashberg theory of phonon-mediatedsuperconductivity in 2D systems in light of recent extensive Monte-Carlostudies of the Holstein model. Conventional wisdom says that Eliashberg theoryis applicable as long as vertex corrections remain small. For small ratio ofthe phonon energy $\Omega_0$ and the Fermi energy $E_F$, this condition issupposed to hold even when the dimensionless electron-phonon coupling $\lambda$is larger than one, i.e., in the strong coupling regime. A comparison betweenvarious quantities computed in the Migdal approximation and those computed byQuantum Monte Carlo prove that this belief is wrong, and we identifyanalytically some of the ways in which this breakdown occurs for various"normal state" properties at $\lambda = \lambda_{cr}$, where $\lambda_{cr} =O(1)$. The breakdown occurs at temperatures high enough that neithersuperconducting nor charge-density wave correlations extend over anysignificant range of distances, so it cannot be associated with the onset of aninstability toward any of the relevant ordered ground-states - rather it isassociated with the local physics of classical bipolaron formation. Still, weshow that certain properties, including the superconducting $T_c$ and thesuperconducting gap structure below $T_c$, can be accurately inferred from thestrong-coupling limit of Eliashberg theory at $\lambda \leq \lambda_{cr}$.

Eliashberg theory of phonon-mediated superconductivity -- when it is valid and how it breaks down