Code embeddings have seen increasing applications in software engineering (SE) research and practice recently. Despite the advances in embedding techniques applied in SE research, one of the main challenges is their generalizability. A recent study finds that code embeddings may not be readily leveraged for the downstream tasks that the embeddings are not particularly trained for. Therefore, in this article, we propose GraphCodeVec , which represents the source code as graphs and leverages the Graph Convolutional Networks to learn more generalizable code embeddings in a task-agnostic manner. The edges in the graph representation are automatically constructed from the paths in the abstract syntax trees, and the nodes from the tokens in the source code. To evaluate the effectiveness of GraphCodeVec , we consider three downstream benchmark tasks (i.e., code comment generation, code authorship identification, and code clones detection) that are used in a prior benchmarking of code embeddings and add three new downstream tasks (i.e., source code classification, logging statements prediction, and software defect prediction), resulting in a total of six downstream tasks that are considered in our evaluation. For each downstream task, we apply the embeddings learned by GraphCodeVec and the embeddings learned from four baseline approaches and compare their respective performance. We find that GraphCodeVec outperforms all the baselines in five out of the six downstream tasks, and its performance is relatively stable across different tasks and datasets. In addition, we perform ablation experiments to understand the impacts of the training context (i.e., the graph context extracted from the abstract syntax trees) and the training model (i.e., the Graph Convolutional Networks) on the effectiveness of the generated embeddings. The results show that both the graph context and the Graph Convolutional Networks can benefit GraphCodeVec in producing high-quality embeddings for the downstream tasks, while the improvement by Graph Convolutional Networks is more robust across different downstream tasks and datasets. Our findings suggest that future research and practice may consider using graph-based deep learning methods to capture the structural information of the source code for SE tasks.