Appears in collection : 7ème Séminaire Itzykson : « Résurgence et quantification »

Divergent series are ubiquitous in quantum theory, and one should learn to live with them and eventually make sense of them. A general framework to address these issues is the theory of resurgence by Ecalle. This requires the introduction of trans-series, or generalized formal series incorporating exponentially small effects, which are invisible in classical asymptotics. I will present the general framework of resurgence in down-to-earth terms, and discuss various examples in the theory of differential equations, Quantum Mechanics and in some (simple) Quantum Field Theories. In the last part of the talk I will summarize recent progress in the application of this program to topological string theory.