  The accurate modelling of the Point Spread Function (PSF) is of paramount
importance in astronomical observations, as it allows for the correction of
distortions and blurring caused by the telescope and atmosphere. PSF modelling
is crucial for accurately measuring celestial objects' properties. The last
decades brought us a steady increase in the power and complexity of
astronomical telescopes and instruments. Upcoming galaxy surveys like Euclid
and LSST will observe an unprecedented amount and quality of data. Modelling
the PSF for these new facilities and surveys requires novel modelling
techniques that can cope with the ever-tightening error requirements. The
purpose of this review is three-fold. First, we introduce the optical
background required for a more physically-motivated PSF modelling and propose
an observational model that can be reused for future developments. Second, we
provide an overview of the different physical contributors of the PSF,
including the optic- and detector-level contributors and the atmosphere. We
expect that the overview will help better understand the modelled effects.
Third, we discuss the different methods for PSF modelling from the parametric
and non-parametric families for ground- and space-based telescopes, with their
advantages and limitations. Validation methods for PSF models are then
addressed, with several metrics related to weak lensing studies discussed in
detail. Finally, we explore current challenges and future directions in PSF
modelling for astronomical telescopes.
