  A general method for constructing sharply $k$-arc-transitive digraphs, i.e.
digraphs that are $k$-arc-transitive but not $(k+1)$-arc-transitive, is
presented. Using our method it is possible to construct both finite and
infinite examples. The infinite examples can have one, two or infinitely many
ends. Among the one-ended examples there are also digraphs that have polynomial
growth.
