Cosmic inflation is a phase of accelerated expansion that occurred in the early universe, during which vacuum quantum fluctuations were amplified by gravitational instability and gave rise to density fluctuations on large scales. These fluctuations constitute the seeds of all cosmological structures, and the validity of this scenario has been confirmed by a wealth of high-precision astrophysical measurements, ranging from the temperature and polarisation fluctuations of the Cosmic Microwave Background, to galaxy and large-scale structure surveys. When primordial inhomogeneities are produced with sufficiently large amplitude in the early universe, they may subsequently collapse into primordial black holes. I will explain why the effect of quantum diffusion during inflation needs to be taken into account in such a case, and how the statistics of cosmological fluctuations can be predicted within the formalism of stochastic inflation. In particular, I will show that quantum diffusion leads to a type of non-Gaussianity that cannot be captured by perturbative parametrisations, and which leaves specific imprints on the statistics of collapsed structures.