In my talk I will explain how higher and infinite derivatives naturally arise in a consideration of quantum properties of general gravity theories. I will construct the most general propagator around Minkowski and (anti)-de Sitter backgrounds proving that an infinite tower of derivatives is inevitable in order to exorcise ghosts. Then I will build on this explaining cosmological implications: modification of a tensor power spectrum, large non-Gaussianities, ideas towards observable primordial gravitational waves and related questions.