It is often said that perturbation theory is insufficient to understand many physical problems, and that non-perturbative effects are needed. It turns out that making this statement precise requires the mathematical framework of resurgence theory, in which perturbative series are extended to so-called trans-series, and non-perturbative effects can be detected by looking at perturbation theory at large orders. In this talk I will first review the basics of the theory of resurgence, and then present some of its recent applications to condensed matter systems. In particular, I will argue that the superconducting gap can be understood in terms of the Borel singularities of the perturbative series in the normal state, and is similar to the renormalon singularity of quantum chromodynamics.