Graph data, such as citation networks, social networks, and transportation networks, exist widely in the real world. Graph neural networks (GNNs) have attracted extensive attention due to their strong expressiveness and excellent performance in a variety of graph analysis applications. However, the excellent performance of GNNs benefits from label data which are difficult to obtain, and complex network models with high computational costs. Knowledge distillation (KD) is introduced into the GNNs to address the labeled data scarcity and high complexity of GNNs. KD is a method of training constructed small models (student models) by soft-label supervision information from larger models (teacher models) to yield better performance and accuracy. Therefore, how to apply the KD technology to graph data has become a research challenge, but there is still a lack of a graph-based KD research review. Aiming at providing a comprehensive overview of KD based on graphs, this study first summarizes the existing studies and fills in the review gap in this field. Specifically, this study first introduces the background knowledge of graph and KD. Then, three types of graph-based knowledge distillation methods are comprehensively summarized, including graph knowledge distillation for deep neural networks (DNNs), graph knowledge distillation for GNNs, and self-KD-based graph knowledge distillation. Furthermore, each type of method is further divided into knowledge distillation methods based on the output layer, the middle layer, and the constructed graph. Subsequently, the design ideas of various graph-based knowledge distillation algorithms are analyzed and compared, and the advantages and disadvantages of the algorithms are concluded with experimental results. In addition, the application of graph-based knowledge distillation in computer vision, natural language processing, recommendation systems, and other fields are also listed. Finally, the development of graph-based knowledge distillation is summarized and prospected. This study also discloses the references related to graph-based knowledge distillation on GitHub. Please refer to https://github.com/liujing1023/Graph-based-Knowledge-Distillation.