The so-called “affine” formulation, already proposed at small strain, has been proved to improve significantly on former treatments such as Hill’s incremental one. This paper reports on an extension of this treatment to finite strain. A linearization procedure, consistent with the one used for small strains, is proposed. This extension has to solve general problems related to the application of classical homogenization schemes to finite strain situations, as well as new ones which are specific to the affine formulation. Preliminary results concerning the tensile stress-strain response of fcc polycrystals are favorably compared with those derived from Hill’s approach.