  The back-reaction of a classical gravitational field interacting with quantum
matter fields is described by the semiclassical Einstein equation, which has
the expectation value of the quantum matter fields stress tensor as a source.
The semiclassical theory may be obtained from the quantum field theory of
gravity interacting with N matter fields in the large N limit. This theory
breaks down when the fields quantum fluctuations are important. Stochastic
gravity goes beyond the semiclassical limit and allows for a systematic and
self-consistent description of the metric fluctuations induced by these quantum
fluctuations. The correlation functions of the metric fluctuations obtained in
stochastic gravity reproduce the correlation functions in the quantum theory to
leading order in an 1/N expansion. Two main applications of stochastic gravity
are discussed. The first, in cosmology, to obtain the spectrum of primordial
metric perturbations induced by the inflaton fluctuations, even beyond the
linear approximation. The second, in black hole physics, to study the
fluctuations of the horizon of an evaporating black hole.
