I will explain some recent exact developments in N=2 SUSY gauge theories on 3-sphere and its deformations. I will begin by the analysis of Killing spinors and their generalization which are necessary to construct SUSY theories on curved spaces. Then I will sketch the exact computation of partition function using SUSY localization and present a general formula. Some applications to the physics of M5-branes will also be discussed.

Yukawa, Kyoto