Using a generally applicable dynamic structural system of equations, we give natural definitions of direct and total structural causality applicable to both structural VARs and recursive structures representing time-series natural experiments. These concepts enable us to forge a previously missing link between Granger (G-) causality and structural causality by showing that, given a corresponding conditional form of exogeneity, G- causality holds if and only if a corresponding form of structural causality holds. Of importance for applications is the structural characterization of finite-order G-causality, which forms the basisfor most empirical work. We show that conditional exogeneity is necessary for valid structural inference and prove that in the absence of structural causality, conditional exogeneity is equivalent to G non-causality. We provide practical new G-causality and conditional exogeneity tests and describe their use in testing for structural causality.