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The increasing complexity of MAPF in complex and crowded environments, however, critically diminishes the effectiveness of existing solutions. In contrast to other studies that have either presented a general overview of the recent advancements in MAPF or extensively reviewed Deep Reinforcement Learning (DRL) within multi-agent system settings independently, our work presented in this review paper focuses on highlighting the integration of DRL-based approaches in MAPF. Moreover, we aim to bridge the current gap in evaluating MAPF solutions by addressing the lack of unified evaluation indicators and providing comprehensive clarification on these indicators. Finally, our paper discusses the potential of model-based DRL as a promising future direction and provides its required foundational understanding to address current challenges in MAPF. 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