Taxonomies, which organize knowledge hierarchically, support various practical web applications such as product navigation in online shopping and user profile tagging on social platforms. Given the continued and rapid emergence of new entities, maintaining a comprehensive taxonomy in a timely manner through human annotation is prohibitively expensive. Therefore, expanding a taxonomy automatically with new entities is essential. Most existing methods for expanding taxonomies encode entities into vector embeddings (i.e., single points). However, we argue that vectors are insufficient to model the “is-a” hierarchy in taxonomy (asymmetrical relation), because two points can only represent pairwise similarity (symmetrical relation). To this end, we propose to project taxonomy entities into boxes (i.e., hyperrectangles). Two boxes can be "contained", "disjoint" and "intersecting", thus naturally representing an asymmetrical taxonomic hierarchy. Upon box embeddings, we propose a novel model BoxTaxo for taxonomy expansion. The core of BoxTaxo is to learn boxes for entities to capture their child-parent hierarchies. To achieve this, BoxTaxo optimizes the box embeddings from a joint view of geometry and probability. BoxTaxo also offers an easy and natural way for inference: examine whether the box of a given new entity is fully enclosed inside the box of a candidate parent from the existing taxonomy. Extensive experiments on two benchmarks demonstrate the effectiveness of BoxTaxo compared to vector based models.