I will explain how primordial black holes can form from
perturbations seeded during inflation and how their abundance can be
calculated in the framework of stochastic inflation. This formalism
incorporates quantum backreaction of the small-wavelength fluctuations
on the large distances dynamics of the Universe. If quantum
corrections are small, the probability distribution of density
fluctuations is well approximated by a Gaussian. If they are large,
the PDF has a different profile with a longer tail and leads to