  This is an introductory level review of recent applications of resurgent
trans-series and Picard-Lefschetz theory to quantum mechanics and quantum field
theory. Resurgence connects local perturbative data with global topological
structure. In quantum mechanical systems, this program provides a constructive
relation between different saddles. For example, in certain cases it has been
shown that all information around the instanton saddle is encoded in
perturbation theory around the perturbative saddle. In quantum field theory,
such as sigma models compactified on a circle, neutral bions provide a
semi-classical interpretation of the elusive IR-renormalon, and fractional kink
instantons lead to the non-perturbatively induced gap, of order of the strong
scale. In the path integral formulation of quantum mechanics, saddles must be
found by solving the holomorphic Newton's equation in the inverted
(holomorphized) potential. Some saddles are complex, multi-valued, and even
singular, but of finite action, and their inclusion is strictly necessary to
prevent inconsistencies. The multi-valued saddles enter either via resurgent
cancellations, or their phase is tied with a hidden topological angle. We
emphasize the importance of the destructive/constructive interference effects
between equally dominant saddles in the Lefschetz thimble decomposition. This
is especially important in the context of the sign problem.
