Let A be a finite automaton. We are concerned with the minimal length of the words that send all states to a unique state (synchronizing words). J. Cerný has conjectured that, if there exists a synchronizing word in A then there exists such a word with length ? (n-1)^2 where n is the number of states of A. The aim of this paper is to present some results about this conjecture, to indicate a generalized approach of the problem and to give a complete solution for a special case.