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During inflation, vacuum quantum fluctuations are amplified and stretched to astrophysical distances. They give rise to fluctuations in the cosmic microwave background (CMB) temperature and polarisation, and to large-scale structures in our universe. They can also trigger the formation of primordial black holes (PBHs). Such objects could provide the progenitors of the recently detected black-hole mergers, and constitute part or all of the dark matter. Their observation would give invaluable access to parts of the inflationary sector that are unconstrained by the CMB. Since PBHs require large inhomogeneities to form, they are produced in scenarios where quantum fluctuations substantially modify the dynamics of the universe. In this habilitation thesis, this "backreaction" effect is investigated using the stochastic inflation formalism, an effective theory for the long-wavelengths of quantum fields during inflation, which can be described in a classical but stochastic way once the small wavelengths have been integrated out. It describes an inflating background that gets randomly corrected by the vacuum quantum fluctuations as they get stretched to large distances. After a brief review of the stochastic inflation formalism, we explain how it can be combined with standard techniques of cosmological perturbation theory (the $δN$ formalism) to provide the full probability density function of curvature perturbations in the presence of non-perturbative quantum diffusion. These results are then applied to PBHs, where we show that quantum diffusion can change the expected abundance by several orders of magnitude. Finally, since inflationary models giving rise to cosmologically relevant PBHs often feature violations of slow roll, the stochastic-$δN$ formalism is generalised to non slow-roll dynamics. We conclude by highlighting several research directions that remain to be explored.