It is a classic result that any normally hyperbolic operator on a globally hyperbolic spacetime admits unique advanced and retarded propagators. With the advent of quantum field theory, a new type of propagator emerges — the Feynman propagator. These propagators are an essential ingredient of quantum field theory and are intimately connected with quantum states. Moreover, they arise naturally in global and spectral analyses on Lorentzian manifolds. In contrast to the advanced and retarded propagators, Feynman propagators are not unique unless the underlying spacetime admits a time-translation symmetry. In this talk, I shall first present a construction of Feynman propagators for a normally hyperbolic operator, satisfying a positivity property based on the idea of microlocalisation which, in a sense, is an efficient tool to connect a first-order pseudodifferential operator of real principal type to a partial derivative. In the last part of this talk, another construction of Feynman propagators for Dirac-type operators will discussed. These results, in turn, provide a novel microlocal construction of Hadamard states of wave (resp. Dirac)-type operators. (Joint work with A. Strohmaier based on arXiv:2012.09767 [math.AP]).