Very often in physics and mathematics we have an asymptotic expansion in small parameter which is divergent because the coefficients have factorial growth an ∼ n! exp(O(n)). The property of resurgence means that the Borel transform ∑n anζn/n! admits an analytic continuation, which allows to reconstruct the exact value of the original divergent series.
I will review the theory of resurgence and resummation for the asymptotic expansions of exponential integrals. I will illustrate the theory by finite-dimensional examples (Airy function, Stirling formula for Γ-function), and infinite-dimensional ones (heat kernel, quantum Chern-Simons theory).