  We present a review on the state-of-the-art of the mathematical framework
known as stochastic inflation, paying special attention to its derivation and
giving references for the readers interested on results coming from the
application of the stochastic framework to different inflationary scenarios,
especially to those of interest for primordial black hole formation. During the
derivation of the stochastic formalism, we will emphasise two aspects in
particular: the difference between the separate universe approach and the true
long wavelength limit of scalar inhomogeneities and the generically
non-Markovian nature of the noises that appear in the stochastic equations.
