In order to study the weakly coupled regime of some given quantum theory we often make use of perturbative expansions of the physical quantities of interest. But such expansions are often divergent, with zero radius of convergence, and defined only as asymptotic series.
In fact, this divergence is connected to the existence of nonperturbative contributions, i.e. instanton effects that cannot be simply captured by a perturbative analysis. The theory of resurgence is a mathematical tool which allows us to effectively study this connection and its consequences. Moreover, it allows us to construct a full non-perturbative solution from perturbative data. In this talk, I will review the essential role of resurgence theory in the description of the analytic solution behind the asymptotic series. I will then re-
late resurgence to the so-called Stokes phenomena and phase transitions using a simple example, and will further discuss some major applications of this construction.