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.
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