Generating functions are a major theme in mathematics which unify many disciplines. It is remarkably common to find a combinatorial generating function which factors as a ratio of products of $q$-integers. Examples of such quotients arise from enumerative combinatorics (e.g. subset sums and $q$-binomials), representation theory (the $q$-Weyl dimension formula), geometry (iterated $\mathbb{P}^1$-bundles), probability (random variable decompositions), commutative algebra (homogeneous systems of parameters), and more. We call such quotients "cyclotomic generating functions" (CGFs) and initiate their general study. This talk will review some of the many known constructions of CGFs and give asymptotic estimates of their coefficients. We will also highlight a range of conjectures and accessible open problems. Joint work in progress with Sara Billey.