Families of N=2 field theories

This is the first article in the collection of reviews “Exact results on N=2 supersymmetric gauge theories”, ed. J. Teschner. It describes how large families of field theories with N=2 supersymmetry can be described by means of Lagrangian formulations, or by compactification from the six-dimensional theory with (2,0) supersymmetry on spaces of the form $M^4\times C$, with C being a Riemann surface. The class of theories that can be obtained in this way is called class $\mathcal{S}$. This description allows us to relate key aspects of the four-dimensional physics of class $\mathcal{S}$ theories to geometric structures on C.