Let G be finite group and let S be a subset of G. We prove a necessary and sufficient condition for the Cayley digraph X ( G , S ) to be primitive when S contains the central elements of G. As an immediate consequence we obtain that a Cayley digraph X ( G , S ) on an Abelian group is primitive if and only if S - 1 S is a generating set for G. Moreover, it is shown that if a Cayley digraph X ( G , S ) on an Abelian group is primitive, then its exponent either is n - 1, [n2], [n2] - 1 or is not exceeding [n3] + 1. Finally, we also characterize those Cayley digraphs on Abelian groups with exponent n - 1, [n2], [n2] - 1. In particular, we generalize a number of well-known results for the primitive circulant matrices.