Recent years have witnessed rapid progress in the use of integrability in characterizing the out-of-equilibrium dynamics of low-dimensional systems such as interacting atomic gases and quantum spin chains. This talk will provide an introduction to these developments, with a particular focus on the Quench Action method. Exact solutions to the interaction turn-on quench in the Lieb-Liniger model and to the Néel-to-XXZ quench in spin chains will be presented. Particular emphasis will be given to interesting open issues and challenges, including the failure of the (local) Generalized Gibbs Ensemble to properly describe post-quench steady-state properties and the necessity to include quasilocal conserved charges to obtain correct answers.