The purpose of the paper is to explain the general theory of coherent functors on the category of finitely generated modules over a noetherian ring \(A\), and to give some applications to cohomology of coherent sheaves on projective spaces. In the special case where \(A\) is a discrete valuation ring, a necessary and sufficient condition for a functor to be coherent is given in terms of commuting with certain inverse limits. Several applications are given: Grothendieck's relative duality theorem is stated in terms of dual coherent functors, the behavior under liaison of the Rao functor of a flat family of space curves is described, and a generalization of a theorem of Horrocks is given.