I will discuss two distinct operator-theoretic settings useful fordescribing (or defining) propagators associated with a scalar Klein-Gordonfield on a Lorentzian manifold $M$. Typically, I will assume that $M$ is globally hyperbolic. Here, the term propagator refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Classical or, especially, Quantum Field Theory.The off-shell setting is based on the Hilbert space $L^2(M)$. It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called out-in Feynman and in-out anti-Feynman propagator. The on-shell setting is based on the Krein space $\sW_\KG$ of solutions of the Klein-Gordon equation. It allows us to define 2-point functions associated to two, possibly distinct Fock states as the Klein-Gordon kernels of projectors onto maximal uniformly positive subspaces of $\sW_\KG$. After a general discussion, I will review a number of examples. I start with static and asymptotically static spacetimes, which are especially well-suited for Quantum Field Theory. (If time permits) then I discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schrodinger operators, de Sitter space and the (universal covering of) anti-de Sitterspace. Based on a joint work with Christian Gass.