Abstract: Graph neural networks (GNNs) are powerful tools for handling graph-structured data, capable of capturing complex relationships and features among nodes. However, the discrete architecture of GNNs leads to numerous challenges in representing graph structures, modeling graph evolution, adapting to irregular data, and managing computational costs. In response to these challenges, neural ordinary differential equations (ODEs) have been introduced as a novel method to address the challenges faced by GNNs, as they can simulate the continuous evolution of system states, providing continuous deep encoding and inference capabilities. However, neural ODEs are designed for Euclidean structured data and cannot directly capture the characteristics of graphs. Therefore, researchers have proposed graph neural ODEs, a new type of architectures that combines neural ODEs with GNNs, which can better adapt to graph-structured data and fully utilize its characteristics. In recent years, research related to graph neural ODEs has delved into various directions of graph machine learning, sparking a new research trend. In this context, we systematically review the relevant research of graph neural ODEs in a timely manner. Firstly, we review the key advantages of GNNs and the challenges they face, and elucidate the theoretical basis and practical significance of introducing neural ODEs and combining them with GNNs. Subsequently, we elaborate on the background and basic concepts of graph neural ODEs, proposing a novel taxonomy, and comprehensively describe some important current methods on the taxonomy. Then, we introduce commonly used verification methods in related research, including downstream tasks and datasets. Furthermore, we delve into the applications of graph neural ODEs in multiple practical fields. Finally, we summarize and prospect the challenges and future development trends of graph neural ODEs.