  The formation and abundance of primordial black holes (PBHs) arising from the
curvature perturbation $\zeta$ is studied. The non-linear relation between
$\zeta$ and the density contrast $\delta$ means that, even when $\zeta$ has an
exactly Gaussian distribution, significant non-Gaussianities affecting PBH
formation must be considered. Numerical simulations are used to investigate the
critical value and the mass of PBHs which form, and peaks theory is used to
calculate the mass fraction of the universe collapsing to form PBHs at the time
of formation. A formalism to calculate the total present day PBH abundance and
mass function is also derived. It is found that the abundance of PBHs is very
sensitive to the non-linear effects, and that the power spectrum
$\mathcal{P}_\zeta$ must be a factor of $\sim2$ larger to produce the same
number of PBHs as the linear model (where the exact value depends on the
critical value for a region to collapse and form a PBH). This also means that
the derived constraints on the small-scale power spectrum from constraints on
the abundance of PBHs are weaker by the same factor.
