Over the past few years, colloidal copper chalcogenide nanocrystals (NCs) have emerged as promising alternatives to conventional Cd and Pb chalcogenide NCs. Owing to their wide size, shape, and composition tunability, Cu chalcogenide NCs hold great promise for several applications, such as photovoltaics, lighting and displays, and biomedical imaging. They also offer characteristics that are unparalleled by Cd and Pb chalcogenide NCs, such as plasmonic properties. Moreover, colloidal Cu chalcogenide NCs have low toxicity, potentially lower costs, and excellent colloidal stability. This makes them attractive materials for the large-scale deployment of inexpensive, sustainable, and environmentally benign solution-processed devices. Nevertheless, the synthesis of colloidal Cu chalcogenide NCs, especially that of ternary and quaternary compositions, has yet to reach the same level of mastery as that available for the prototypical Cd chalcogenide based NCs. This review provides a concise overview of this rapidly advancing field, sketching the state of the art and highlighting the key challenges. We discuss recent developments in the synthesis of size-, shape-, and composition-controlled NCs of Cu chalcogenides, with emphasis in strategies to circumvent the limitations arising from the need to precisely balance the reactivities of multiple precursors in synthesizing ternary and quaternary compositions. In this respect, we show that topotactic cation-exchange reactions are a promising alternative route to complex multinary Cu chalcogenide NCs and hetero-NCs, which are not attainable by conventional routes. The properties and potential applications of Cu chalcogenide NCs and hetero-NCs are also addressed.