This volume contains fourteen survey lectures: three on algebraic geometry, two on differential geometry, one on the Poincaré conjecture, one on dynamic systems, one on number theory, one on the fundamentals lemma, one on the André-Oort conjecture, one on quadratic forms, one on algebraic topology, one on mathematical physics, and one on probabilities. As tradition dictates, all of them contain a discussion of important achievements (in this case in the period 2004-2005). For instance, the article devoted to number theory reports on the Green-Tao result on arithmetic progressions in the sequence of primes. The book also contains contributions on the Mumford conjecture (following the work of Madsen and Weiss), a proof of the Parisi formula by Guerra and Talagrand, on ‘Formes quadratiques et cycles algébriques’ (following Rost, Voevodsky, Vishik, Karpenko and others), and so on. As one of the most important sources of reports on major achievements in contemporary mathematics, the volume will find a prominent place on every shelf.