In the past 30 years, Quantum Field Theory and Integrable Systems have been developing hand in hand. To cite a few examples: the classical algebraic integrable systems in Seiberg-Witten theory, the integrable massive deformations of two dimensional conformal theories, the recent relation between the vacua of supersymmetric theories and quantum integrable systems, such as spin chains, many-body systems, and even more abstract ones, such as the Hitchin system. Some time ago, it was shown that the quantization of the integrable systems appearing in the Seiberg-Witten geometry is beautifully described by a deformation of the four-dimensional, N=2 supersymmetric quantum field theory.  Recently, this was related to another deep connection between gauge theories and the geometric Langlands program, as well as to the connection between the instanton partition functions and conformal blocks of two-dimensional theories, such as Liouville theory.

This circle of ideas has surprising connections to other important areas of current research, such as wall-crossing phenomena, the thermodynamic Bethe ansatz, scattering amplitudes in super-Yang-Mills, black hole entropy counting, and tt*-systems, etc.