  We consider rigid supersymmetric theories in four-dimensional Riemannian spin
manifolds. We build the Lagrangian directly in Euclidean signature from the
outset, keeping track of potential boundary terms. We reformulate the
conditions for supersymmetry as a set of conditions on the torsion classes of a
suitable SU(2) or trivial G-structure. We illustrate the formalism with a
number of examples including supersymmetric backgrounds with non-vanishing Weyl
tensor.
