My work fits into the research topics of the BELL research group “Fundamental Problems in quantum physics” in Genoa (Group IV) and concerns the fundamental aspects of a theory which describes the interplay between gravity and quantum physics. More precisely, the framework of my research project is the quantum field theory on curved spacetimes and the general relativity, namely the so-called semiclassical gravity. The starting point of my work is the study of the mathematical and physical conditions which insure the analyticity of the expansion of a two-point correlation functions, for a flat and curved spacetime. Then, the topic of my research is the semiclassical analysis of the backreaction effects of a quantum matter field upon the spacetime geometry, which are governed by the semiclassical Einstein equation, in cosmological models and black hole physics.