**Abstract** : We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and
call such a colouring an out-colouring. The problem of deciding whether a given digraph has
an out-colouring with only two colours (called a 2-out-colouring) is NP-complete. We show that
for every choice of positive integers r, k there exists a k-strong bipartite tournament which needs
at least r colours in every out-colouring. Our main results are on tournaments and semicomplete
digraphs. We prove that, except for the Paley tournament P7, every strong semicomplete digraph
of minimum out-degree at least 3 has a 2-out-colouring. Furthermore, we show that every semi-
complete digraph on at least 7 vertices has a 2-out-colouring if and only if it has a balanced such
colouring, that is, the difference between the number of vertices that receive colour 1 and colour 2
is at most one. In the second half of the paper we consider the generalization of 2-out-colourings
to vertex partitions (V1,V2) of a digraph D so that each of the three digraphs induced by re-
spectively, the vertices of V1, the vertices of V2 and all arcs between V1 and V2 have minimum
out-degree k for a prescribed integer k ≥ 1. Using probabilistic arguments we prove that there
exists an absolute positive constant c so that every semicomplete digraph of minimum out-degree √
at least 2k + c k has such a partition. This is tight up to the value of c.