Abstract: Solvable interesting problems in physics are very rare. Almost all nontrivial problems require approximations, the most common of which is perturbation theory. We are all familiar with the basic idea of perturbation theory, but it holds some surprising secrets, which can be understood physically. These secrets also introduce some novel concepts from a new mathematical formalism called "resurgent asymptotics". The talk is largely non-technical, concentrating on the key physical ideas and examples, illustrated with applications in quantum mechanics and quantum field theory.