  This is the 5th article in the collection of reviews "Exact results on N=2
supersymmetric gauge theories", ed. J. Teschner. We review the supersymmetric
localization of $\mathcal{N}=2$ theories on curved backgrounds in four
dimensions using $\mathcal{N}=2$ supergravity and generalised conformal Killing
spinors. We review some known backgrounds and give examples of new geometries
such as local $T^2$-bundle fibrations. We discuss in detail a topological
four-sphere with generic $T^2$-invariant metric.
