Recent years have seen an upsurge of interest in deformations of two-dimensional sigma-models which preserve classical integrability. Integrability is known to offer powerful techniques for solving such models exactly, even in complex scenarios such as at strong coupling. This talk introduces classical integrability, and the role played by worldsheet dualities in the development of a large family of integrable deformations. The second part of the talk focuses on the application of these deformations within the AdS/CFT correspondence, in order to obtain exact methods for addressing gauge and gravity theories with reduced Noether (super)symmetries. However, current "AdS/CFT integrability" methods are mostly restricted to the undeformed, maximally (super)symmetric instances. To enhance their applicability to a broader range of theoretical models, the concept of “twisted” AdS/CFT integrability is introduced, specifically targeting the “Jordanian” subclass of integrable deformations. Recent and ongoing work in this area will be discussed.