  In this paper, based on lectures by the authors at the May 2015 workshop {\it
Resurgence, Physics and Numbers}, at the Centro di Ricerca Matematica Ennio De
Giorgio of the Scuola Normale Superiore in Pisa, we explain the origin of
resurgent trans-series in the Mathieu equation spectral problem, using uniform
WKB and all-orders (exact) WKB. Exact quantization conditions naturally arise,
and their expansion in the weak coupling regime produces resurgent trans-series
expressions which exhibit precise relations between different instanton
sectors. Indeed, the perturbative expansion encodes all orders of the
multi-instanton expansion, an explicit realization of the general concept of
"resurgence". We also discuss the transition from weak to strong coupling, an
explicit realization of "instanton condensation".
