Various recently developed connections between supersymmetric Yang-Mills theories in four dimensions and two-dimensional integrable systems serve as crucial ingredients in improving our understanding of the AdS/CFT correspondence. In this review, we highlight some connections between superconformal four-dimensional Yang-Mills theory and various integrable systems. In particular, we focus on the role of Yangian symmetries in studying the gauge theory dual of closed string excitations. We also briefly review how the gauge theory connects to Calogero models and open quantum spin chains through the study of the gauge theory duals of D3 branes and open strings ending on them. This invited review is based on a seminar given at the Institute of Advanced Study, Princeton.