  In stellar astrophysics, the technique of population synthesis has been
successfully used for several decades. For planets, it is in contrast still a
young method which only became important in recent years because of the rapid
increase of the number of known extrasolar planets, and the associated growth
of statistical observational constraints. With planetary population synthesis,
the theory of planet formation and evolution can be put to the test against
these constraints. In this review of planetary population synthesis, we first
briefly list key observational constraints. Then, the work flow in the method
and its two main components are presented, namely global end-to-end models that
predict planetary system properties directly from protoplanetary disk
properties and probability distributions for these initial conditions. An
overview of various population synthesis models in the literature is given. The
sub-models for the physical processes considered in global models are
described: the evolution of the protoplanetary disk, the planets' accretion of
solids and gas, orbital migration, and N-body interactions among concurrently
growing protoplanets. Next, typical population synthesis results are
illustrated in the form of new syntheses obtained with the latest generation of
the Bern model. Planetary formation tracks, the distribution of planets in the
mass-distance and radius-distance plane, the planetary mass function, and the
distributions of planetary radii, semimajor axes, and luminosities are shown,
linked to underlying physical processes, and compared with their observational
counterparts. We finish by highlighting the most important predictions made by
population synthesis models and discuss the lessons learned from these
predictions - both those later observationally confirmed and those rejected.
