In traditional graph learning tasks, such as node classification, the learning is carried out in a closed-world setting where the number of classes and their training samples are provided to help train models, and the learning goal is to correctly classify unlabeled nodes into classes already known. In reality, due to limited labeling capability and dynamic evolving of networks, some nodes in the networks may not belong to any existing/seen classes, and therefore cannot be correctly classified by closed-world learning algorithms. In this paper, we propose a new open-world graph learning paradigm, where the learning goal is to not only classify nodes belonging to seen classes into correct groups, but also classify nodes not belonging to existing classes to an unseen class. The essential challenge of the open-world graph learning is that (1) unseen class has no labeled samples, and may exist in an arbitrary form different from existing seen classes; and (2) both graph feature learning and prediction should differentiate whether a node may belong to an existing/seen class or an unseen class. To tackle the challenges, we propose an uncertain node representation learning approach, using constrained variational graph autoencoder networks, where the label loss and class uncertainty loss constraints are used to ensure that the node representation learning are sensitive to unseen class. As a result, the node embedding features are denoted by distributions, instead of deterministic feature vectors. By using a sampling process to generate multiple versions of feature vectors, we are able to test the certainty of a node belonging to seen classes, and automatically determine a threshold to reject nodes not belonging to seen classes as unseen class nodes. Experiments on real-world networks demonstrate the algorithm performance, comparing to baselines. Case studies and ablation analysis also show the rationale of our design for open-world graph learning.