(received ; accepted ) PACS. 71.38+i -Polarons and electron-phonon interactions. PACS. 63.20Kr -Phonon-electron and phonon-phonon interactions.Abstract. -We study the crossover from quasi free electron to small polaron in the Holstein model for a single electron by means of both exact and self-consistent calculations in one dimension and on an infinite coordination lattice. We show that the crossover occurs when both strong coupling (λ > 1) and multiphonon (α 2 > 1) conditions are fulfilled leading to different relevant coupling constants (λ) in adiabatic and (α 2 ) anti adiabatic region of the parameters space. We also show that the self-consistent calculations obtained by including the first electron-phonon vertex correction give accurate results in a sizeable region of the phase diagram well separated from the polaronic crossover.Recent optical measurements of the insulating parent compounds of the high-temperature superconductors [1] show the presence of polaronic carriers, and evidence for strong electronphonon (el-ph) coupling effects has been given also for the colossal magnetoresistance manganites [2] and Nickel compounds [3]. These findings underline the necessity of a clear theoretical description of electron-phonon coupled system and more specifically of the constraints for the existence of the small polaron ground state. This state, characterized by strong local electron-lattice correlation, is definitively a non-perturbative phenomenon, and cannot be described by simple summation of the perturbative series such as the one which defines the Migdal-Eliashberg (ME) theory [4,5].The aim of this work is to provide a detailed study of the crossover which occurs at intermediate electron-lattice couplings from quasi-free electron to small polaron ground state. We also study the role of the lattice dimensionality and compare exact results with selfconsistent theories.