Learning continuous-time dynamics on complex networks is crucial for understanding, predicting and controlling complex systems in science and engineering. However, this task is very challenging due to the combinatorial complexities in the structures of high dimensional systems, their elusive continuous-time nonlinear dynamics, and their structural-dynamic dependencies. To address these challenges, we propose to combine Ordinary Differential Equation Systems (ODEs) and Graph Neural Networks (GNNs) to learn continuous-time dynamics on complex networks in a data-driven manner. We model differential equation systems by GNNs. Instead of mapping through a discrete number of neural layers in the forward process, we integrate GNN layers over continuous time numerically, leading to capturing continuous-time dynamics on graphs. Our model can be interpreted as a Continuous-time GNN model or a Graph Neural ODEs model. Our model can be utilized for continuous-time network dynamics prediction, structured sequence prediction (a regularly-sampled case), and node semi-supervised classification tasks (a one-snapshot case) in a unified framework. We validate our model by extensive experiments in the above three scenarios. The promising experimental results demonstrate our model's capability of jointly capturing the structure and dynamics of complex systems in a unified framework.