The Feynman propagator is at the heart of quantum field theory. However, in quantum field theory in curved spacetimes, no locally covariant notion of a distinguished Feynman propagator exists. Instead, often a distinguished class of Feynman propagators is considered, which share a common parametrix. Nevertheless, certain classes of spacetimes possess distinguished Feynman propagators. First, I will give an in-depth introduction to propagators (Green functions) on curved spacetimes and their role in quantum field theory. In particular, I will highlight the importance of the so-called Hadamard states – an appropriate generalization of the Poincaré invariant vacuum state. Then, I will show that the free Klein–Gordon field on asymptotically static spacetimes comes equipped with a natural Feynman propagator (albeit globally constructed and generally not related to a state). Finally, I will argue that this Feynman propagator is closely related to the question of the self-adjointness of the Klein–Gordon operator on L2(spacetime) and the boundary value of its resolvent.