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The purpose of our article is to extend the classical notion of Fre´chet differentiability to multifunctions. To this end we define the notion of affinity for multifunctions and study the basic properties of affine multifunctions. Then using affine multifunctions as local approximations and the Hausdorff distance for defining an approximation mode, we introduce the notion of Fre´chet differentiability for multifunctions mapping points of a finite-dimensional normed space to compact convex subsets of another finite-dimensional normed space. We characterize Fre´chet differentiability of multifunctions through the differentiable properties of their support functions and discuss the relationship of our notion of differentiability with other ones which were studied by Blagodatskikh (Blagodatskikh, V.I., 1984, Maximum principle for differential inclusions. Trudy Matematicheskogo Instituta AN SSSR, 166, 23-43 (in Russian)), Rubinov (Rubinov, A.M., 1985, The conjugate derivative of a multivalued mapping and differentiability of the maximum function under connected constraints. Sibirskii Matematicheskii Zhurnal, 26(3), 147-155 (in Russian)), Tyurin (Tyurin, Yu. N., 1965, A mathematical formulation of a simplified model of industrial planning.

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