The vector spaces of generalized theta functions on smooth curves have been studied quite actively from multiple perspectives. For instance, their dimensions are computed by the celebrated Verlinde formula, which drew much interest in 90s. These spaces fit together into a vector bundle over the moduli space of curves, which extends to the Deligne-Mumford compactification, and it has recently been discovered, due primarily to work of Fakhruddin, that these vector bundles offer an intriguing view of the divisor theory and birational geometry of these moduli spaces. In this talk, I'll discuss some of the interest in these topics, such as Mumford's question from the 70s and the F-conjecture, and I'll show how conformal blocks have been used in the past few years to relate these questions to some elegant classical geometry, such as the Torelli map, the Gale transform, and the geometry of rational normal curves.