Background/Objectives: The healthcare sector is rapidly growing, aiming to promote health, provide treatment, and enhance well-being. This paper focuses on the organ donation and transplantation system, a vital aspect of healthcare. It offers a comprehensive review of challenges in global organ donation and transplantation, highlighting issues of fairness and transparency, and compares centralized architecture-based models and blockchain-based decentralized models. Methods: This work reviews 370 publications from 2016 to 2023 on organ donation management systems. Out of these, 85 publications met the inclusion criteria, including 67 journal articles, 2 doctoral theses, and 16 conference papers. About 50.6% of these publications focus on global challenges in the system. Additionally, 12.9% of the publications examine centralized architecture-based models, and 36.5% of the publications explore blockchain-based decentralized models. Results: Concerns about organ trafficking, illicit trade, system distrust, and unethical allocation are highlighted, with a lack of transparency as the primary catalyst in organ donation and transplantation. It has been observed that centralized architecture-based models use technologies such as Python, Java, SQL, and Android Technology but face data storage issues. In contrast, blockchain-based decentralized models, mainly using Ethereum and a subset on Hyperledger Fabric, benefit from decentralized data storage, ensure transparency, and address these concerns efficiently. Conclusions: It has been observed that blockchain technology-based models are the better option for organ donation management systems. Further, suggestions for future directions for researchers in the field of organ donation management systems have been presented.