  We discuss various aspects of rigid supersymmetry within minimal N=1
off-shell supergravity using the old and new minimal formulations both in
Lorentzian and Euclidean signatures. In particular, we construct all rigid
supersymmetry backgrounds with a hypersurface orthogonal Killing vector. In the
Lorentzian signature we show that AdS_4 provides a rigid supersymmetric
background in both formulations albeit with different amounts of preserved
supersymmetry. In the Euclidean signature we find new backgrounds of the
old-minimal supergravity, including squashed four-spheres and a half-BPS
version of flat space.
