  The aim of these notes is to acquaint the reader with important objects in
complex algebraic geometry: K3 surfaces and their higher-dimensional analogs,
hyperk\"ahler manifolds. These manifolds are interesting from several points of
view: dynamical (some have interesting automorphism groups), arithmetical
(although we will not say anything on this aspect of the theory), and
geometric. It is also one of those rare cases where the Torelli theorem allows
for a powerful link between the geometry of these manifolds and lattice theory.
  We do not prove all the results that we state. Our aim is more to provide,
for specific families of hyperk\"ahler manifolds (which are projective
deformations of punctual Hilbert schemes of K3 surfaces), a panorama of results
about projective embeddings, automorphisms, moduli spaces, period maps and
domains, rather than a complete reference guide. These results are mostly not
new, except perhaps those of Appendix B (written with E. Macr\`i), where we
give an explicit description of the image of the period map for these polarized
manifolds.
