Recent years have demonstrated that integrability can be used to compute many physical properties of experimentally-accessible magnetic or cold atomic systems. Besides their rich equilibrium dynamics, such systems also host relaxation and equilibration behaviour which cannot be simply described by traditional textbook methods, and can lead to long-lived non-thermal equilibrium states. This colloquium will provide an overview of this area, highlighting ideas and lessons of import for researchers in other fields and/or using methods complementary to integrability.