Graph neural networks (GNNs) are recognized as a significant methodology for handling graph-structure data. However, with the increasing prevalence of learning scenarios involving multiple graphs, traditional GNNs mostly overlook the relationships between nodes across different graphs, mainly due to their limitation of traditional message passing within each graph. In this paper, we propose a novel GNN architecture called cross-graph interaction networks (GInterNet) to enable inter-graph message passing. Specifically, we develop a cross-graph topology construction module to uncover and learn the potential topologies between nodes across different graphs. Furthermore, we establish inter-graph message passing based on the learned cross-graph topologies, achieving cross-graph interaction by aggregating information from different graphs. Finally, we employ cross-graph construction functions involving the relationships between contextual information and cross-graph topology structure to iteratively update the cross-graph topologies. Different to existing related approaches, GInterNet is designed as a cross-graph interaction paradigm for inter-graph message passing. It enables multi-graph interaction during the message passing process. Additionally, it is a plug-and-play framework that can be easily embedded into other models. We evaluate its performance in semi-supervised and unsupervised learning scenarios involving multiple graphs. A detailed theoretical analysis and extensive experiment results have shown that GInterNet improves the performance and robustness of the base models.