Conditions de stabilit\'e et g\'eom\'etrie birationnelle [d'apr\`es Bridgeland, Bayer-Macr\`i, ...]

We survey some aspects of stability conditions both in general and on the derived category of coherent sheaves on a surface, with applications to the birational geometry of certain holomorphic symplectic varieties.
Comment: Texte de l'expos\'e 1147 du s\'eminaire Bourbaki (juin 2018). Text of talk 1147 of the Bourbaki seminar (june 2018), 47 pages, in French