We consider face-to-face partitions of bounded polytopes into convex polytopes in d for arbitrary d ≥ 1 and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed d+ 1. Partitions of polyhedra in 3 into pentahedra and hexahedra are 5- and 6-colourable, respectively. We show that the above numbers are attainable, i.e., in general, they cannot be reduced.