The Hall algebra is an invariant of an abelian category C whose
multiplication comes from "counting extensions in C." These algebras
typically have interesting representation theory, and they have found
applications in knot theory, mathematical physics, combinatorics, and
more. In this talk we discuss some of the history of Hall algebras,
old and new, and then give a conjectural description of the Hall
algebra of the Fukaya category of a topological surface.
(Joint work with B. Cooper.)