Explore the intricacies of braided fusion categories in this comprehensive lecture by Dmitri Nikshych. Delve into the algebraic theory of semisimple braided tensor categories, their structure, and classification. Learn about their natural occurrence in quantum group representation theory and their role in describing non-commutative or "quantum" symmetries. Discover applications in topological quantum field theories, Galois theory for Jones-von Neumann subfactors, and topological phases of matter. Cover key topics including basic properties and examples, centralizers, (de)-equivariantizations, core of braided fusion categories, gauging, Picard groups, and Witt equivalence of non-degenerate categories. Suitable for graduate students, this 50-minute talk is part of the Workshop on Quantum symmetries: Tensor Categories, Topological quantum field theories, Vertex algebras held at the Centre de recherches mathématiques (CRM).