Let K:P→Tbe a fixed functor. A criterion is given for a functor M':T→Vto be a (right) Kan extension along K of some functor M:P→V.The functors M having a given M' as Kan extension are, in general, classified by continuous functors (VP)o→V.We introduce a notion of system of imprimitivity, generalizing that of Mackey; when the shape category of K is codense in the systems of imprimitivity classify the functors H having M' as Kan extension. As a special case one obtains Mackey's Imprimitivity Theorem for finite groups.