  We present a non-perturbative method for calculating the abundance of
primordial black holes given an arbitrary one-point probability distribution
function for the primordial curvature perturbation, $P(\zeta)$. A
non-perturbative method is essential when considering non-Gaussianities that
cannot be treated using a conventional perturbative expansion. To determine the
full statistics of the density field, we relate $\zeta$ to a Gaussian field by
equating the cumulative distribution functions. We consider two examples: a
specific local-type non-Gaussian distribution arising from ultra slow roll
models, and a general piecewise model for $P(\zeta)$ with an exponential tail.
We demonstrate that the enhancement of primordial black hole formation is due
to the intermediate regime, rather than the far tail. We also show that
non-Gaussianity can have a significant impact on the shape of the primordial
black hole mass distribution.
