  This paper begins with a brief survey of the period prior to and soon after
the creation of the theory of vertex operator algebras (VOAs). This survey is
intended to highlight some of the important developments leading to the
creation of VOA theory. The paper then proceeds to describe progress made in
the field of VOAs in the last 15 years which is based on fruitful analogies and
connections between VOAs and commutative algebras. First, there are several
functors from VOAs to commutative algebras that allow methods from commutative
algebra to be used to solve VOA problems. To illustrate this, we present a
method for describing orbifolds and cosets using methods of classical invariant
theory. This was essential in the recent solution of a conjecture of Gaiotto
and Rap\v{c}\'ak that is of current interest in physics. We also recast some
old conjectures in the subject in terms of commutative algebra and give some
generalizations of these conjectures. We also give an overview of the theory of
topological VOAs (TVOAs), with applications to BRST cohomology theory and
conformal string theory, based on work in the 90's. We construct a functor from
TVOAs to Batalin-Vilkovisky algebras -- supercommutative algebras equipped with
a certain odd Poisson structure realized by a second order differential
operator -- and present a number of interesting applications. This paper is
based in part on the lecture given by the first author at the Harvard CMSA
Math-Science Literature Lecture Series on May 22, 2020.
