R 4 terms and d = 4 supergravity

We start by a concise yet thorough revision of four-dimensional super-space supergravity. We present curved superspace geometry, for arbitrary N , including torsion, curvature and Bianchi identities. We motivate the choice of torsion constraints. We then consider the particular cases of N = 1, 2. In both cases we show how Poincaré supergravity can be obtained from conformal supergravity. We see how to obtain the different versions of the Poincaré off-shell theory, with distinct compensating multiplets and sets of auxiliary fields. For those versions of N = 1, 2 supergravities known as " old minimal " , we present the solutions to the Bianchi identities, their field content and we show how to write superspace actions for these theories and their extensions using chiral densities and chiral projectors. As concrete applications, we study the supersymmetrization of the two possible R 4 terms in d = 4, which are both required as string corrections to supergravity.