Summary: We first recall the classical theory of theta functions: in modern terms, they are sections of a line bundle on a complex torus. We discuss a fundamental example of such a complex torus, the Jacobian \(JC\) of an algebraic curve \(C\), which parametrizes topologically trivial line bundles on \(C\). Then we define generalized theta functions by replacing \(JC\) by a moduli space of higher rank vector bundles. We discuss the similarities and differences with the classical theory, the known results and the open problems.
For the entire collection see [Zbl 1279.00039].