  Weak lensing of galaxies and CMB photons through the large-scale structure of
the Universe is one of the most promising cosmological probes with upcoming
experiments dedicated to its measurements such as Euclid/LSST and CMB Stage 4
experiments. With increasingly precise measurements, there is a dire need for
accurate theoretical predictions. In this work, we focus on higher order
statistics of the weak lensing convergence field, namely its cumulants such as
skewness and kurtosis and its one-point probability distribution (PDF), and we
quantify using perturbation theory the corrections coming from post-Born
effects, meaning beyond the straight-line and independent lenses
approximations. At first order, two such corrections arise: lens-lens couplings
and geodesic deviation. Though the corrections are small for low source
redshifts (below a few percents) and therefore for galaxy lensing, they become
important at higher redshifts, notably in the context of CMB lensing, where the
non-gaussianities computed from tree-order perturbation theory are found to be
of the same order as the signal itself. We include these post-Born corrections
on the skewness into a prediction for the one-point convergence PDF obtained
with large deviation theory and successfully test these results against
numerical simulations. The modelled PDF is indeed shown to perform better than
the percent for apertures above ~ 10 arcminutes and typically in the three
sigmas region around the mean.
