  This paper is a survey of results and problems related to the following
question: is it true that if G is a tournament with sufficiently large
chromatic number, then G has two vertex-disjoint subtournaments A,B, both with
large chromatic number, such that all edges between them are directed from A to
B? We describe what we know about this question, and report some progress on
several other related questions, on tournament colouring and domination.
