The goal of this talk is to give a friendly introduction to several questions related to the remarkable fact that linear differential equations on the complex plane are actually of topological nature. This result, known as the Riemann-Hilbert correspondence, says that a complex linear ODE is fully characterized by its monodromy, a topological object which describes what happens when one follows the solutions of the equation along a loop around one of its singularities.

An interesting consequence of this arises when we consider deformations of linear ODEs: looking for deformations which keep the monodromy constant amounts to an integrable system of nonlinear PDEs. Many equations ubiquitous in mathematical physics, for example all Painlevé equations, arise in this way. These PDEs themselves admit a topological version, consisting of the action of a discrete group on the space parametrizing all possible monodromy data.

I will try to explain the main ideas involved in this rich picture, and, if time permits, also say a few words about the case of irregular singularities and some of my work.