This paper presents a comprehensive review of the last two decades of research on Kidney Exchange Programs (KEPs), systematically categorizing and classifying key contributions to provide readers with a structured understanding of advancements in the field. The review highlights the evolution of KEP methodologies and lays the foundation for our contribution. We propose three mathematical models aimed at improving both the quantity and quality of kidney transplants. Model 1 maximizes the number of transplants by focusing on compatibility based on blood type and PRA, without additional constraints. Model 2 introduces a minimum Human Leukocyte Antigen (HLA) compatibility threshold to enhance transplant quality, though this leads to fewer matches. Model 3 extends the problem to a Multi-Agent Kidney Exchange Program (MKEP), pooling incompatible donor-recipient pairs across multiple agents, resulting in a higher number of successful transplants while ensuring fairness across agents. Sensitivity analyses demonstrate trade-offs between transplant quantity and quality, with Model 3 striking the optimal balance by leveraging multi-agent collaboration to improve both the number and quality of transplants. These findings underscore the potential benefits of more integrated kidney exchange systems.