Growth outlines the behavior of a class of inventories whose weight and size increase during their storage. Fast-growing young animals such as broiler chickens, ducks, and calves are examples of this class, prevalent in many food industries. Despite their presence in various food supply chains, studying these items in the context of inventory control, supply chain, and operations management is still in its early stage. This study provides the essential grounds of the knowledge in this area and further reviews the related research developments to provide the researchers with clear insights into the structure of the current literature, its main gaps, and promising directions for future research. In order to guarantee the comprehensiveness of our review, we have incorporated a systematic search procedure to identify the relevant literature. Through this procedure, 23 papers were identified that were used in our further analysis throughout the paper. Our investigations show that the literature on growing inventories is very confined, and this body of the literature gets even more limited when it comes to the problems in the context of the supply chain. Our findings introduce the topic as a promising direction in inventory and supply chain management with high potential for insightful future research works.