Graph neural networks (GNNs), as a branch of deep learning in non-Euclidean space, perform particularly well in various tasks that process graph structure data. With the rapid accumulation of biological network data, GNNs have also become an important tool in bioinformatics. In this research, a systematic survey of GNNs and their advances in bioinformatics is presented from multiple perspectives. We first introduce some commonly used GNN models and their basic principles. Then, three representative tasks are proposed based on the three levels of structural information that can be learned by GNNs: node classification, link prediction, and graph generation. Meanwhile, according to the specific applications for various omics data, we categorize and discuss the related studies in three aspects: disease prediction, drug discovery, and biomedical imaging. Based on the analysis, we provide an outlook on the shortcomings of current studies and point out their developing prospect. Although GNNs have achieved excellent results in many biological tasks at present, they still face challenges in terms of low-quality data processing, methodology, and interpretability and have a long road ahead. We believe that GNNs are potentially an excellent method that solves various biological problems in bioinformatics research.