Given a linear ordinary differential equation in one complexvariable z, e.g. a "Schrodinger equation" (d2 / dz2 + P(z)) f(z) = 0,one would like to understand the solutions as well as possible. Oneconcrete question, much studied over the last 100 years, is: what is themonodromy of the solutions when z goes around a loop? Recently a newscheme for solving this problem has been discovered, which gives muchmore precise information than was previously available. This new schemealso revealed surprising connections to various areas of mathematics,such as the combinatorics of cluster algebras, the theory of enumerativeinvariants (generalized Donaldson-Thomas invariants of 3-Calabi-Yaucategories), 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 willdescribe some of these developments, focusing on concrete examples.