Integrable deformations of conformal field theory

Two-dimensional conformal field theories may have deformations away from the critical point which preserve an infinite number of commuting conserved charges, that is, they are integrable. The generating functions of these conserved charges can often be arranged to satisfy functional equations, the so-called Y-system. In this talk I describe a first principles approach to finding such integrable deformations and the corresponding Y-systems starting from a detailed understanding of the conformal field theory and its line defects.