It is known that every complete Pick Hilbert function space is a quotient of the Drury-Arveson space. Likewise, every multiplier algebra of such a space is a quotient of the multiplier algebra of the Drury-Arveson space. Motivated by this fact, researchers have been trying to understand the structure of these spaces and algebras in complex geometric terms. During the last decade, several beautiful and striking discoveries have been made, as well as some unexpected applications to function theory and complex geometry. In this talk, I will review some of the highlights of this classification program and then turn to some recent quantitative results.