I will discuss how to classify (up to discrete identifications) all
rigid 4D N=2 supersymmetric backgrounds in both Lorentzian and Euclidean
signatures that preserve eight real supercharges. These include
backgrounds such as warped S_3×R, warped AdS_3×R, and AdS_2×S^2, as well
as some more exotic geometries. Time permitting, I will also address
how to construct all supersymmetric two-derivative actions involving
hypermultiplets and vector multiplets in these backgrounds.