  We review the superspace technique to determine supersymmetric spacetimes in
the framework of off-shell formulations for supergravity in diverse dimensions
using the case of 3D N=2 supergravity theories as an illustrative example. This
geometric formalism has several advantages over other approaches advocated in
the last four years. Firstly, the infinitesimal isometry transformations of a
given curved superspace form, by construction, a finite-dimensional Lie
superalgebra, with its odd part corresponding to the rigid supersymmetry
transformations. Secondly, the generalised Killing spinor equation, which must
be obeyed by the supersymmetry parameters, is a consequence of the more
fundamental superfield Killing equation. Thirdly, general rigid supersymmetric
theories on a curved spacetime are readily constructed in superspace by making
use of the known off-shell supergravity-matter couplings and restricting them
to the background chosen. It is the superspace techniques which make it
possible to generate arbitrary off-shell supergravity-matter couplings.
Fourthly, all maximally supersymmetric Lorentzian spaces correspond to those
off-shell supergravity backgrounds for which the Grassmann-odd components of
the superspace torsion and curvature tensors vanish, while the Grassmann-even
components of these tensors are annihilated by the spinor derivatives.
