  Stochastic semiclassical gravity is a theory for the interaction of gravity
with quantum matter fields which goes beyond the semiclassical limit. The
theory predicts stochastic fluctuations of the classical gravitational field
induced by the quantum fluctuations of the stress energy tensor of the matter
fields. Here we use an axiomatic approach to introduce the Einstein-Langevin
equations as the consistent set of dynamical equations for a first order
perturbative correction to semiclassical gravity and review their main
features. We then describe the application of the theory in a simple chaotic
inflationary model, where the fluctuations of the inflaton field induce
stochastic fluctuations in the gravitational field. The correlation functions
for these gravitational fluctuations lead to an almost Harrison-Zel'dovich
scale invariant spectrum at large scales, in agreement with the standard
theories for structure formation. A summary of recent results and other
applications of the theory is also given.
