The study of the geometry and topology of Riemann surfaces has captured the interest of geometers for decades. They lie in the intersection of many areas of mathematics: we can aim to understand representations of their fundamental groups, we can attempt to classify differential equations on them, or perhaps study their algebraic vector bundles. It is also tempting to try to classify all compact Riemann surfaces themselves. Moduli theory provides a way to understand many of these classification problems, by turning our focus to the geometry of a parameter space. These parameter spaces, called moduli spaces, provide us in turn with a lot of examples of complex varieties with rich geometric features. In this talk I will discuss some recent techniques developed to construct such moduli spaces and study their geometry. With time permitting, I will also try to explain what it means to count vector bundles on compact Riemann surfaces, and why such counts are given by combinations of certain special values of transcendental functions.