Cu-based shape memory alloys belong to one important class of functional alloys, presenting shape memory effect and superelasticity due to their reversible martensitic transformation. Although they have been extensively studied since the middle of the last century, there are still many challenges to be solved. In the last decades, these alloys were extensively studied regarding new compositions, processing routes, phase transformation, mechanical and functional properties. Aspects of the thermoelastic phase transformation have been described using thermodynamic and thermo-mechanical studies, while the role of metallurgical features (such as grain size and morphology, ordering, precipitates and second phases) have been described mainly by phenomenological approach. In this sense this review discusses the advances in the general fundamentals of Cu-based shape memory alloys, the recent developments in processing routes, compositions, and applications in the last years. Graphical abstract