This article reviews the microlocal construction of Feynman propagators for normally hyperbolic operators acting on vector bundles over globally hyperbolic spacetimes and its consequences. It is shown that for normally hyperbolic operators that are selfadjoint with respect to a hermitian bundle metric, the Feynman propagators can be constructed to satisfy a positivity property that reflects the existence of Hadamard states in quantum field theory on curved spacetimes. We also give a more direct construction of the Feynman propagator for the Dirac operator on a globally hyperbolic spacetime. Even though the natural bundle metric on spinors is not positive-definite, in this case we can give a direct microlocal construction of a Feynman propagator that satisfies positivity.