  The formalism to determine (conformal) isometries of a given curved
superspace was elaborated almost two decades ago in the context of the old
minimal formulation for N=1 supergravity in four dimensions (4D). This
formalism is universal, for it may readily be generalized to supersymmetric
backgrounds associated with any supergravity theory formulated in superspace.
In particular, it has already been used to construct rigid supersymmetric field
theories in 5D N=1, 4D N=2 and 3D (p,q) anti-de Sitter superspaces. In the last
two years, there have appeared a number of publications devoted to the
construction of supersymmetric backgrounds in off-shell 4D N=1 supergravity
theories using component field considerations. Here we demonstrate how to read
off the key results of these recent publications from the more general
superspace approach developed in the 1990s. We also present a universal
superspace setting to construct supersymmetric backgrounds, which is applicable
to any of the known off-shell formulations for N=1 supergravity. This approach
is based on the realizations of the new minimal and non-minimal supergravity
theories as super-Weyl invariant couplings of the old minimal supergravity to
certain conformal compensators.
