Let \(G\) be a finite group. Especially, we denote the cyclic group of order \(n (\geq 3)\) by \(Z_n\). A finite simple directed graph \(H\) is called a directed graphical regular representation (abbreviated as DRR) of \(G\) if (i) the automorphism group of \(H\) is isomorphic to \(G\) and (ii) whenever \((u,v)\) is an ordered pair consisting of vertices of \(H\), then there is exactly one automorphism \(f\) of \(H\) such that \(f(u) = v\). If, in addition, the outdegree of any vertex of \(H\) equals \(k\), then we say that \(H\) is a directed graphical \(k\)-regular representation (shortly \(k\)-DRR) of \(G\). Two theorems are stated. Theorem 1 asserts that \(Z_n\) has a \(k\)-DRR if and only if \(0 < k < n - 1\). The 2-DRRs of \(Z_n\) are described in Theorem 2.