Electric vehicles and portable electronic devices frequently employ lithium-ion batteries (LIBs) with high power densities. A large number of LIBs are being used as a result of their applications’ explosive growth in recent years. An appropriate response is required to the emerging threat that spent LIBs with heavy metals and other hazardous materials pose, to the environment and public health. Metals like lithium (Li), nickel (Ni), cobalt (Co), and manganese (Mn) can be extracted from LIB waste. These end-of-life LIBs have the option of being recycled, which can ultimately lower the cost of producing LIBs by preventing the pollution of harmful components, conserving and preserving natural resources. The recycling processes for lithium-ion batteries mainly include hydrometallurgy and pyrometallurgy among which this review primarily focuses on the most recent technological advancements in hydrometallurgical leaching processes that use organic acids (such as citric acid, ascorbic acid, D, L-malic acid, and tartaric acid) to recover critical metals from waste LIBs. The benefits of organic acids over inorganic acids have subsequently been covered as well.