This volume contains 15 articles based on survey talks at the Bourbaki
seminar in Paris in the academic year 2005/06. They cover a wide range of topics: Julia sets of positive measure, (non)-regularity of solutions of partial differential equations, singular integral operators, rigidity of Bernoulli actions, non-projective Kähler manifolds, p-adic analytic geometry, compactifications of moduli spaces, classification of p-divisible groups, Arakelov geometry, the Serre modularity conjecture, prime number gaps, the perfect graph theorem and representations of Lie superalgebras.