  We start by a concise yet thorough revision of four-dimensional superspace
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 Poincare supergravity can be obtained from conformal supergravity. We
see how to obtain the different versions of the Poincare 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. We conclude by discussing
possible applications of these results to open problems on black holes in
string theory.
