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This is a detailed survey which mainly presents the Pinkham-Feller way. Iadded some new points to the first version [V2] and I suppressed "Examples"devoted to Gamma, Fr\'echet and Weibull laws. Theorem 2 is a bit more general(no assumption of density: this answers a question of T. Hill). Section 10 isnew and devoted to an argument (Poincar\'e, Fewster) about the effect of highfrequencies oscillations. Maybe many works, many efforts, have been devoted tothe study of a sufficient condition of poor value: see Sections 7 and 11. Thefinal Section gives some suggestions.

The Benford phenomenon for random variables. Discussion of Feller's way