AbstractLet G = (V, A) be a digraph with a root r, and suppose that each arc a of G has integers b(a) and b'(a) with b′(a)⩾b(a)⩾0. In this paper we give necessary and sufficient conditions for the existence of k r-arborescences in G such that they cover each arc a at least b(a) and at most b'(a) times and each of them contains exactly one arc leaving r.The main result of this paper contains as special cases a number of packing and covering theorems concerning arborescences