Driven by breakthroughs in experimental and theoretical techniques, the study of nonequilibrium quantum physics is a rapidly expanding field with many exciting new developments. Among the manifold ways the topic can be investigated, one-dimensional systems provide a particularly fine platform. The trifecta of strongly correlated physics, powerful theoretical techniques, and experimental viability have resulted in a flurry of research activity over the past decade or so. In this review, we explore the nonequilibrium aspects of one-dimensional systems that are integrable. Through a number of illustrative examples, we discuss nonequilibrium phenomena that arise in such models, the role played by integrability, and the consequences these have for more generic systems.