Learning from source code usually requires a large amount of labeled data. Despite the possible scarcity of labeled data, the trained model is highly task-specific and lacks transferability to different tasks. In this work, we present effective pre-training strategies on top of a novel graph-based code representation, to produce universal representations for code. Specifically, our graph-based representation captures important semantics between code elements (e.g., control flow and data flow). We pre-train graph neural networks on the representation to extract universal code properties. The pre-trained model then enables the possibility of fine-tuning to support various downstream applications. We evaluate our model on two real-world datasets – spanning over 30M Java methods and 770K Python methods. Through visualization, we reveal discriminative properties in our universal code representation. By comparing multiple benchmarks, we demonstrate that the proposed framework achieves state-of-the-art results on method name prediction and code graph link prediction.