The subject of this thesis is the coupling of quantum fields to a classical gravitational background in a semiclassical fashion. It contains a thorough introduction into quantum field theory on curved spacetime with a focus on the stress-energy tensor and the semiclassical Einstein equation. Basic notions of differential geometry, topology, functional and microlocal analysis, causality and general relativity will be summarised, and the algebraic approach to QFT on curved spacetime will be reviewed. Apart from these foundations, the original research of the author and his collaborators will be presented: Together with Fewster, the author studied the up and down structure of permutations using their decomposition into so-called atomic permutations. The relevance of these results to this thesis is their application in the calculation of the moments of quadratic quantum fields. In a work with Pinamonti, the author showed the local and global existence of solutions to the semiclassical Einstein equation in flat cosmological spacetimes coupled to a scalar field by solving simultaneously for the quantum state and the Hubble function in an integral-functional equation. The theorem is proved with a fixed-point theorem using the continuous functional differentiability and boundedness of the integral kernel of the integral-functional equation. In another work with Pinamonti the author proposed an extension of the semiclassical Einstein equations which couples the moments of a stochastic Einstein tensor to the moments of the quantum stress-energy tensor. In a toy model of a Newtonianly perturbed exponentially expanding spacetime it is shown that the quantum fluctuations of the stress-energy tensor induce an almost scale-invariant power spectrum for the perturbation potential and that non-Gaussianties arise naturally.