  The present paper is the [slightly expanded] text of our talk at the
Conference "Advances in Group Theory and Applications" at Porto Cesareo in June
2011. Our main results assert that [elementary] Chevalley groups very rarely
have finite commutator width. The reason is that they have very few
commutators, in fact, commutators have finite width in elementary generators.
We discuss also the background, bounded elementary generation, methods of
proof, relative analogues of these results, some positive results, and possible
generalisations.
