In order to study the weakly coupled regimeof some given quantum theory we often make use of perturbative expansions ofthe physical quantities of interest. But such expansions are often divergent,with zero radius of convergence, and defined only as asymptotic series. Infact, this divergence is connected to the existence of nonperturbativecontributions, i.e. instanton effects that cannot be simply captured by aperturbative analysis. The theory of resurgence is a mathematical tool whichallows us to effectively study this connection and its consequences. Moreover,it allows us to construct a full nonperturbative solution from perturbativedata. The essential role of resurgence theory become evident when one enforcessome physical conditions, such as reality requirements, to the analyticsolution behind the asymptotic series. In this talk, we will see that enforcingreality for our physical quantities requires not only the asymptotic series butalso all other instanton sectors due to the non-linearity of most problems: allproperly described via resurgence theory. I will then relate resurgence to theso-called Stokes phenomena and phase transitions, and will further discuss somemajor applications of this construction, with emphasis in obtaining a fullpicture of the phase diagram of matrix models through the study of thenonperturbative sectors in these models.