This summer school is organized jointly by the Vietnam Institute for Advanced Study in Mathematics (Hanoi, Vietnam) and the International Center for Theoretical Physics (Trieste, Italy), and in partnership with the Clay Mathematics Institute. There will be five 4-hour minicourses given by F. Bonahon, F.Costantino, S. Garoufalidis, I. Irmer, and C. Manolescu. Manolescu is the Clay lecturer. The minicourses are accompanied by brainstorm sessions. The goal is to introduce graduate students and young researchers to basics of quantum topology and its connection to many branches of mathematics, including low-dimensional topology, representations of quantum groups, combinatorics, and in particular hyperbolic geometry. It will also introduce the audience to applications of quantum topology to various areas of mathematics.