Abstract. I discuss various aspects of rigid supersymmetry within minimal N = 1 off-shell supergravity using the old and new minimal formulations in both Lorentzian and Euclidean signatures. In particular, I present a construction of all rigid supersymmetry backgrounds with a hypersurface orthogonal Killing vector. I will also examine the dependence of four-dimensional Euclidean N = 1 partition functions on coupling constants. In particular, I shall focus on backgrounds without R-symmetry, which arise in the rigid limit of old minimal supergravity. Based on a systematic analysis of supersymmetric invariants, I will argue that localization is not possible for backgrounds without R-symmetry.