Taxonomies consist of machine-interpretable semantics and provide valuable knowledge for many web applications. For example, online retailers (e.g., Amazon and eBay) use taxonomies for product recommendation, and web search engines (e.g., Google and Bing) leverage taxonomies to enhance query understanding. Enormous efforts have been made on constructing taxonomies either manually or semi-automatically. However, with the fast-growing volume of web content, existing taxonomies will become outdated and fail to capture emerging knowledge. Therefore, in many applications, dynamic expansions of an existing taxonomy are in great demand. In this paper, we study how to expand an existing taxonomy by adding a set of new concepts. We propose a novel self-supervised framework, named TaxoExpan, which automatically generates a set of pairs from the existing taxonomy as training data. Using such self-supervision data, TaxoExpan learns a model to predict whether a query concept is the direct hyponym of an anchor concept. We develop two innovative techniques in TaxoExpan: (1) a position-enhanced graph neural network that encodes the local structure of an anchor concept in the existing taxonomy, and (2) a noise-robust training objective that enables the learned model to be insensitive to the label noise in the self-supervision data. Extensive experiments on three large-scale datasets from different domains demonstrate both the effectiveness and the efficiency of TaxoExpan for taxonomy expansion.