Appears in collection : 12e Séminaire Itzykson : problèmes spectraux quantiques en physique mathématique

We discuss the relation between Painlevé equations, supersymmetric gauge theories and the Liouville conformal field theory. As an application, we show how one can solve the connection problem for Heun functions in terms of explicit combinatorial formulae arising from supersymmetric localisation. We also present work in progress on the extension of the correspondence to quantum Painlevé equations.