  We construct rigid supersymmetric theories for interacting vector and tensor
multiplets on six-dimensional Riemannian spin manifolds. Analyzing the Killing
spinor equations, we derive the constraints on these theories. To this end, we
reformulate the conditions for supersymmetry as a set of necessary and
sufficient conditions on the geometry. The formalism is illustrated with a
number of examples, including manifolds that are hermitian, strong Kaehler with
torsion. As an application, we show that the path integral of pure super
Yang-Mills theory defined on a Calabi-Yau threefold M_6 localizes on stable
holomorphic bundles over M_6.
