  Stochastic semiclassical gravity of the 90's is a theory naturally evolved
from semiclassical gravity of the 70's and 80's. It improves on the
semiclassical Einstein equation with source given by the expectation value of
the stress-energy tensor of quantum matter fields in curved spacetimes by
incorporating an additional source due to their fluctuations. In stochastic
semiclassical gravity the main object of interest is the noise kernel, the
vacuum expectation value of the (operator-valued) stress-energy bi-tensor, and
the centerpiece is the (stochastic) Einstein-Langevin equation. We describe
this new theory via two approaches: the axiomatic and the functional. The
axiomatic approach is useful to see the structure of the theory from the
framework of semiclassical gravity. The functional approach uses the
Feynman-Vernon influence functional and the Schwinger-Keldysh close-time-path
effective action methods which are convenient for computations. It also brings
out the open systems concepts and the statistical and stochastic contents of
the theory such as dissipation, fluctuations, noise and decoherence. We then
describe the application of stochastic gravity to the backreaction problems in
cosmology and black hole physics. Intended as a first introduction to this
subject, this article places more emphasis on pedagogy than completeness.
