  We investigate the representation theory of finite sets. The correspondence
functors are the functors from the category of finite sets and correspondences
to the category of k-modules, where k is a commutative ring. They have various
specific properties which do not hold for other types of func-tors. In
particular, if k is a field and if F is a correspondence functor, then F is
finitely generated if and only if the dimension of F (X) grows exponentially in
terms of the cardinality of the finite set X. Moreover, in such a case, F has
actually finite length. Also, if k is noetherian, then any subfunctor of a
finitely generated functor is finitely generated.
