Teichm\"uller spaces and Torelli theorems for hyperk\"ahler manifolds

Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperk"ahlerian manifolds as stated by Verbitsky. The initial purpose of this document (which was prepared for a seminar talk) was to extract the correct statement and to give a short proof of it. We also revisit a few of its consequences, some of which are given new (shorter) proofs.