  We study superconformal and supersymmetric theories on Euclidean four- and
three-manifolds with a view toward holographic applications. Preserved
supersymmetry for asymptotically locally AdS solutions implies the existence of
a (charged) "conformal Killing spinor" on the boundary. We study the geometry
behind the existence of such spinors. We show in particular that, in dimension
four, they exist on any complex manifold. This implies that a superconformal
theory has at least one supercharge on any such space, if we allow for a
background field (in general complex) for the R-symmetry. We also show that
this is actually true for any supersymmetric theory with an R-symmetry. We also
analyze the three-dimensional case and provide examples of supersymmetric
theories on Sasaki spaces.
