In this review, we investigate the question of backreaction in different approaches to cosmological perturbation theory, and with a special focus on quantum theoretical aspects. By backreaction we refer here to the effects of matter field or cosmological inhomogeneities on the homogeneous dynamical background degrees of freedom of cosmology. We begin with an overview of classical cosmological backreaction which is ideally suited for physical situations in the late time Universe. We then proceed backwards in time, considering semiclassical approaches such as semiclassical or stochastic (semiclassical) gravity which take quantum effects of the perturbations into account. Finally, we review approaches to backreaction in quantum cosmology that should apply to the very early Universe where classical and semiclassical approximations break down. The main focus is on a recently proposed implementation of backreaction in quantum cosmology using a Born–Oppenheimer inspired method.