Integrability has emerged as a powerful method for obtaining exact non-perturbative results in four-dimensional gauge theories, complementary to lattice, supersymmetric localisation, and the conformal bootstrap. In planar N=4 super Yang-Mills it provides analytic control over the spectrum of anomalous dimensions of local operators at any value of the coupling, currently the only known method for quantising strings in curved Ramond-Ramond backgrounds, and a non-perturbative window onto non-local light-ray operators with direct connections to BFKL physics in QCD. I will review recent developments in the separation-of-variables programme, which promises to extend integrability from the spectrum to structure constants, and discuss applications to lower-dimensional holographic systems.