This chapter gives a very brief overview of the rich theory of Killing spinors describing Bär's correspondence which connects Sasaki-Einstein geometry to Berger's holonomy theorem. It is then shown how weak holonomy G2 and nearly Kähler structures relate to 3-Sasakian 7-manifolds and Sasaki-Einstein 5-manifolds. The chapter ends with a brief discussion of how Sasakian geometry naturally appears in various supersymmetric physical theories such as supergravity theory, superstring theory, and M-theory. In particular, Sasaki-Einstein 5-manifolds play a crucial role in the AdS/CFT duality conjecture.
Keywords: AdS/CFT conjecture, Bär's correspondence, Dirac operator, Killing spinors, M-theory, nearly Kähler manifolds, weak holonomy G2 manifolds, supersymmetry, supergravity