Background: This review addresses the emerging field of automated packing cells, which lies at the intersection of robotics and packing problems. Integrating these two fields is critical for optimizing logistics and e-commerce operations. The current literature focuses on packing problems or specific robotic applications without addressing their integration. Methods: To bridge this gap, we conducted a comprehensive review of 46 relevant studies, analyzing various dimensions, including the components of robotic packing cells, the types of packing problems, the solution approaches, and performance comparisons. Results: Our review reveals a significant trend towards addressing online packing problems, which reflects the dynamic nature of logistics operations where item information is often incomplete. We also identify several research gaps, such as the need for standardized terminologies, comprehensive methodologies, and the consideration of real-world constraints in robotic algorithms. Conclusions: This review uniquely integrates insights from robotics and packing problems, providing a structured framework for future research. It highlights the importance of considering practical robotic constraints. It proposes a research structure that enhances the reproducibility and comparability of results in real-world scenarios. By doing so, we aim to guide future research efforts and facilitate the development of more robust and practical automated packing systems.