Deep learning, especially graph convolutional networks (GCNs), has been widely applied in various scenarios. Particularly in the field of medical image processing, the research on GCNs have continued to make breakthroughs and has been successfully applied to various tasks, such as medical image segmentation, as well as disease detection, localization, classification and diagnosis. GCNs have demonstrated the capacity to autonomously learn latent disease features from vast medical image datasets. Their potential value and enhanced capabilities in prediction, analysis, and decision-making in perioperative medical imaging have become evident. In recent years, GCNs have rapidly emerged as a research focus in the realm of medical image analysis. First, this review provides a concise overview of the development from convolutional neural networks to GCNs, delineating their algorithmic foundations and network structures. Subsequently, the diverse applications of GCNs in perioperative medical image processing are extensively reviewed, including medical image segmentation, image reconstruction, disease prediction, lesion detection and localization, disease classification and diagnosis, and surgical intervention. Finally, this review discusses the prevailing challenges and offers insights into future research directions for the utilization of GCN methods in the medical field.