  We examine the shape of the primordial black hole mass distribution arising
from a peak in the primordial power spectrum. In light of improvements to the
modelling, we revisit the claim that the effects of critical collapse produce a
distribution that is not described by the commonly assumed lognormal, showing
that this conclusion remains valid, particularly for narrow peaks where the
shape of the mass distribution is insensitive to the peak properties and
critical collapse determines a minimum width. We propose some alternative
models that may better describe the shape, both for the narrow peak case and
for much broader peaks where the effect of the peak shape is significant. We
highlight the skew-lognormal and a generalised model motivated by the physics
of critical collapse as the best of these possible alternatives. These models
can be used as an accurate and fast approximation to the numerically calculated
mass distribution, allowing for efficient implementation in an MCMC analysis.
We advocate the use of one of these two models instead of the lognormal with
sufficiently accurate data, such as future LIGO-Virgo observations, or when
considering strongly mass dependent constraints on the PBH abundance.
