This paper provides a comprehensive survey of geometric graph neural networks (GNNs), focusing on their data structures, models, and applications. Geometric graphs are a special type of graph with geometric features, which are crucial for modeling scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries such as translations, rotations, and reflections, which are not effectively handled by traditional GNNs. To address this, researchers have developed geometric GNNs with invariant or equivariant properties to better capture the geometry and topology of geometric graphs. The paper introduces the necessary mathematical preliminaries, including group theory and the concepts of equivariance and invariance. It then defines geometric graphs as a universal data structure for representing scientific data, such as molecules, proteins, and crystals. The paper categorizes existing geometric GNNs into invariant GNNs and equivariant GNNs, with the latter further divided into scalarization-based models and high-degree steerable models. It also discusses geometric graph transformers, which leverage group representation theory to generalize invariant scalars and equivariant vectors. The paper summarizes the applications of geometric GNNs in various domains, including molecular property prediction, protein-ligand docking, and antibody design. It highlights the effectiveness of geometric GNNs in these tasks, demonstrating superior performance compared to traditional methods. The paper also discusses the challenges and future directions of geometric GNNs, including the need for more expressive models and efficient training methods. The paper concludes with a discussion of future prospects and research directions in geometric GNNs, emphasizing the importance of geometric information in modeling scientific systems. It also provides a Github repository that collects references, datasets, codes, benchmarks, and other resources related to geometric GNNs. Overall, the paper provides a comprehensive overview of geometric GNNs, their theoretical foundations, and their applications in various scientific domains.This paper provides a comprehensive survey of geometric graph neural networks (GNNs), focusing on their data structures, models, and applications. Geometric graphs are a special type of graph with geometric features, which are crucial for modeling scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries such as translations, rotations, and reflections, which are not effectively handled by traditional GNNs. To address this, researchers have developed geometric GNNs with invariant or equivariant properties to better capture the geometry and topology of geometric graphs. The paper introduces the necessary mathematical preliminaries, including group theory and the concepts of equivariance and invariance. It then defines geometric graphs as a universal data structure for representing scientific data, such as molecules, proteins, and crystals. The paper categorizes existing geometric GNNs into invariant GNNs and equivariant GNNs, with the latter further divided into scalarization-based models and high-degree steerable models. It also discusses geometric graph transformers, which leverage group representation theory to generalize invariant scalars and equivariant vectors. The paper summarizes the applications of geometric GNNs in various domains, including molecular property prediction, protein-ligand docking, and antibody design. It highlights the effectiveness of geometric GNNs in these tasks, demonstrating superior performance compared to traditional methods. The paper also discusses the challenges and future directions of geometric GNNs, including the need for more expressive models and efficient training methods. The paper concludes with a discussion of future prospects and research directions in geometric GNNs, emphasizing the importance of geometric information in modeling scientific systems. It also provides a Github repository that collects references, datasets, codes, benchmarks, and other resources related to geometric GNNs. Overall, the paper provides a comprehensive overview of geometric GNNs, their theoretical foundations, and their applications in various scientific domains.