  We review the group-geometric approach to supergravity theories, in the
perspective of recent developments and applications. Usual diffeomorphisms,
gauge symmetries and supersymmetries are unified as superdiffeomorphisms in a
supergroup manifold. Integration on supermanifolds is briefly revisited, and
used as a tool to provide a bridge between component and superspace actions. As
an illustration of the constructive techniques, the cases of $d=3,4$ off-shell
supergravities and $d=5$ Chern-Simons supergravity are discussed in detail. A
cursory account of $d=10+2$ supergravity is also included. We recall a
covariant canonical formalism, well adapted to theories described by
Lagrangians $d$-forms, that allows to define a form hamiltonian and to recast
constrained hamiltonian systems in a covariant form language. Finally, group
geometry and properties of spinors and gamma matrices in $d=s+t$ dimensions are
summarized in Appendices.
