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We start by a concise yet thorough revision of four-dimensional superspacesupergravity. We present curved superspace geometry, for arbitrary N, includingtorsion, curvature and Bianchi identities. We motivate the choice of torsionconstraints. We then consider the particular cases of N=1,2. In both cases weshow how Poincare supergravity can be obtained from conformal supergravity. Wesee how to obtain the different versions of the Poincare off-shell theory, withdistinct compensating multiplets and sets of auxiliary fields. For thoseversions of N=1,2 supergravities known as "old minimal", we present thesolutions to the Bianchi identities, their field content and we show how towrite superspace actions for these theories and their extensions using chiraldensities and chiral projectors. As concrete applications, we study thesupersymmetrization of the two possible R^4 terms in d=4, which are bothrequired as string corrections to supergravity. We conclude by discussingpossible applications of these results to open problems on black holes instring theory.

Higher-order string effective actions and off-shell d=4 supergravity