  Primordial Black Holes (PBH's) can form in the early Universe from the
collapse of large density fluctuations. Tight observational limits on their
abundance constrain the amplitude of the primordial fluctuations on very small
scales which can not otherwise be constrained, with PBH's only forming from the
extremely rare large fluctuations. The number of PBH's formed is therefore
sensitive to small changes in the shape of the tail of the fluctuation
distribution, which itself depends on the amount of non-Gaussianity present. We
study, for the first time, how quadratic and cubic local non-Gaussianity of
arbitrary size (parameterised by f_nl and g_nl respectively) affects the PBH
abundance and the resulting constraints on the amplitude of the fluctuations on
very small scales. Intriguingly we find that even non-linearity parameters of
order unity have a significant impact on the PBH abundance. The sign of the
non-Gaussianity is particularly important, with the constraint on the allowed
fluctuation amplitude tightening by an order of magnitude as f_nl changes from
just -0.5 to 0.5. We find that if PBH's are observed in the future, then
regardless of the amplitude of the fluctuations, non-negligible negative f_nl
would be ruled out. Finally we show that g_nl can have an even larger effect on
the number of PBH's formed than f_nl.
