Lithium-ion batteries (LIBs) can play a crucial role in the decarbonization process that is being tackled worldwide; millions of electric vehicles are already provided with or are directly powered by LIBs, and a large number of them will flood the markets within the next 8–10 years. Proper disposal strategies are required, and sustainable and environmental impacts need to be considered. Despite still finding little applicability in the industrial field, recycling could become one of the most sustainable options to handle the end of life of LIBs. This review reports on the most recent advances in sustainable processing for spent LIB recycling that is needed to improve the LIB value chain, with a special focus on green leaching technologies for Co-based cathodes. Specifically, we provide the main state of the art for sustainable LIB recycling processes, focusing on the pretreatment of spent LIBs; we report on Life Cycle Assessment (LCA) studies on the usage of acids, including mineral as well as organic ones; and summarize the recent innovation for the green recovery of valuable metals from spent LIBs, including electrochemical methods. The advantage of using green leaching agents, such as organic acids, which represent a valuable option towards more sustainable recycling processes, is also discussed. Organic acids can, indeed, reduce the economic, chemical, and environmental impacts of LIBs since post-treatments are avoided. Furthermore, existing challenges are identified herein, and suggestions for improving the effectiveness of recycling are defined.