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Résumé. -On décrit les foncteurs polynomiaux, des groupes abéliens libres vers les groupes abéliens, comme des diagrammes de groupes abéliens dont on explicite les relations. Abstract (Polynomial functors and nonlinear Mackey functors)Polynomial functors from free abelian groups to abelian groups are described explicitely in the form of diagrams of abelian groups, that are maps between the crosseffects of the polynomial functor which satisfy a list of relations. The key is to use an appropriate notion of Mackey functor from the category of finite sets and surjections.Soient Ab la catégorie des groupes abéliens, et F (Z) sa sous-catégorie pleine dont les objets sont les groupes abéliens libres de type fini. On sait que la caté-gorie des foncteurs additifs de F (Z) vers Ab estéquivalenteà la catégorie Ab.

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We initiate the study of the cohomology of (strict polynomial) bifunctors by introducing the foundational formalism, establishing numerous properties in analogy with the cohomology of functors, and providing computational techniques. Since one of the initial motivations for the study of functor cohomology was the determination of H *(GL (k), S*(gℓ) ⊗ Λ*(gℓ)), we keep this challenging example in mind as we achieve numerous computations which illustrate our methods.