Given a linear ordinary differential equation in one complex variable z, e.g. a “Schrodinger equation” (d^2 / dz^2 + P(z)) f(z) = 0, one would like to understand the solutions as well as possible. One concrete question, much studied over the last 100 years, is: what is the monodromy of the solutions when z goes around a loop? Recently a new scheme for solving this problem has been discovered, which gives much more precise information than was previously available. This new scheme also revealed surprising connections to various areas of mathematics, such as the combinatorics of cluster algebras, the theory of enumerative invariants (generalized Donaldson-Thomas invariants of 3-Calabi-Yau categories), and the geometry of trajectories of quadratic differentials (and higher analogues). This scheme had its origin in the problem of “BPS state counting” in supersymmetric quantum field theory. I will describe some of these developments, focusing on concrete examples.