In 1997, M. Kontsevich proved that every Poisson manifold admits a formal quantization, canonical up to a suitable equivalence. He thus solved a longstanding problem in mathematical physics. His proof also opened up new horizons for many areas of mathematics like Lie theory, quantum group theory, deformation theory, operads, and their links to knot theory, number theory and the theory of motives. This volume is divided into three parts and one appendix devoted to the geometry of configuration spaces. The first part presents the main result on deformation quantization and includes a description of the Tamarkin approach to it. The second part shows the relevance of the Kontsevich theorem to Lie theory. The third part explains ideas from topological string theory that inspired the Kontsevich proof.