A directed Cayley graph X is called a digraphical regular representation (DRR) of a group G if the automorphism group of X acts regularly on X . Let S be a finite generating set of the infinite cyclic group Z. We show that a directed Cayley graph X (Z, S) is a DRR of Z if and only if S = S −1 . If X (Z, S) is not a DRR we show that Aut (X (Z, S)) = D ∞ . As a general result we prove that a Cayley graph X of a finitely generated torsion-free nilpotent group N is a DRR if and only if no non-trivial automorphism of N of finite order leaves the generating set invariant.