  Complete Pick algebras - these are, roughly, the multiplier algebras in which
Pick's interpolation theorem holds true - have been the focus of much research
in the last twenty years or so. All (irreducible) complete Pick algebras may be
realized concretely as the algebras obtained by restricting multipliers on
Drury-Arveson space to a subvariety of the unit ball; to be precise: every
irreducible complete Pick algebra has the form $M_V = \{f|_V : f \in M_d\}$,
where $M_d$ denotes the multiplier algebra of the Drury-Arveson space $H^2_d$,
and $V$ is the joint zero set of some functions in $M_d$. In recent years
several works were devoted to the classification of complete Pick algebras in
terms of the complex geometry of the varieties with which they are associated.
The purpose of this survey is to give an account of this research in a
comprehensive and unified way. We describe the array of tools and methods that
were developed for this program, and take the opportunity to clarify, improve,
and correct some parts of the literature.
