  We summarize interactions between vertex operator algebras and number theory
through the lens of Zhu theory. The paper begins by recalling basic facts on
vertex operator algebras (VOAs) and modular forms, and then explains Zhu's
theorem on characters of VOAs in a slightly new form. We then axiomatize the
desirable properties of modular forms that have played a role in Zhu's theorem
and related classification results of VOAs. After this we summarize known
classification results in rank two, emphasizing the geometric theory of
vector-valued modular forms as a means for simplifying the discussion. We
conclude by summarizing some known examples, and by providing some new
examples, in higher ranks. In particular, the paper contains a number of
potential character vectors that could plausibly correspond to a VOA, but such
that the existence of a corresponding hypothetical VOA is presently unknown.
