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Quantum Field Theory on curved spacetimes is one of the most fundmental formalisms of theoretical physics. It might be expected that one cannot say many new facts about its foundation. Surprisingly, it seems that some of its fundamental mathematical structures have been clarified only recently (joint work with D.Siemssen). One of such elements of this structure is the notion of the (in-out) Feynman propagator. In this talk, will explain this notion in asymptotically stationary and stable spacetimes. I will also explain how the Feynman propagator is related to the essential self-adjointness of the Klein-Gordon operator.