Graphs are widely used to model the complex relationships among entities. As a powerful tool for graph analytics, graph neural networks (GNNs) have recently gained wide attention due to its end-to-end processing capabilities. With the proliferation of cloud computing, it is increasingly popular to deploy the services of complex and resource-intensive model training and inference in the cloud due to its prominent benefits. However, GNN training and inference services, if deployed in the cloud, will raise critical privacy concerns about the information-rich and proprietary graph data (and the resulting model). While there has been some work on secure neural network training and inference, they all focus on convolutional neural networks handling images and text rather than complex graph data with rich structural information. In this paper, we design, implement, and evaluate SecGNN, the first system supporting privacy-preserving GNN training and inference services in the cloud. SecGNN is built from a synergy of insights on lightweight cryptography and machine learning techniques. We deeply examine the procedure of GNN training and inference, and devise a series of corresponding secure customized protocols to support the holistic computation. Extensive experiments demonstrate that SecGNN achieves comparable plaintext training and inference accuracy, with promising performance.