  We investigate the complexity of generalizations of colourings (acyclic
colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for
the class of transitive digraphs. Even though transitive digraphs are nicely
structured, many problems are intractable, and their complexity turns out to be
difficult to classify. We present some motivational results and several open
problems.
