Hitchin systems in N=2 field theory

This is the second article in the collection of reviews “Exact results on N=2 supersymmetric gauge theories”, ed. J. Teschner. The space of vacua of class $\mathcal{S}$ theories on $R^3\times S^1$ can be identified as the moduli space of solutions to the self-duality equations in two dimensions on Riemann surfaces studied by Hitchin. This fact plays a fundamental role for recent studies of the spectrum of BPS states in class $\mathcal{S}$ theories, and it is related to the integrable structure underlying Seiberg-Witten theory of theories of class $\mathcal{S}$. This article reviews important aspects of the role of the Hitchin system for the infrared physics of class $\mathcal{S}$ theories.