Quantum group symmetry in CFT

I discuss applications of a hidden $U_q({sl}_2)$ symmetry in CFT with central charge $c \leq 1$ (focusing on the generic, semisimple case, withcirrational). This symmetry provides a systematic method for solving Belavin-Polyakov-Zamolodchikov PDE systems, and in partic-ular for explicit calculation of the asymptotics and monodromy properties of the solutions.Using a quantum Schur-Weyl duality, one can understand solution spaces of such PDE systems in a detailed way. The solutions, in turn, are useful both for CFT questions and forrigorous understanding of the connections of 2D CFT with critical models of statistical physics.