Higher structures in Kontsevich's deformation quantization

To a smooth manifold one can associate the Lie algebras of multi-vector fields and multi-differential operators, where one can encode classical data (Poisson structures) and quantum data (star products). Relating these two led Kontsevich to his famous Formality Theorem that established the deformation quantization of Poisson manifolds. In this talk, after a gentle introduction to the topic, I intend to describe a more modern approach that considers natural actions of the (framed) little discs operads which induce richer Batalin-Vilkovisky (BV) algebra structures on the above Lie algebras. I will show how one can obtain a "homotopy BV" version of Kontsevich formality theorem and will explore applications of this result.