Taxonomies are important knowledge ontologies that underpin numerous applications on a daily basis, but many taxonomies used in practice suffer from the low coverage issue. We study the taxonomy expansion problem, which aims to expand existing taxonomies with new concept terms. We propose a self-supervised taxonomy expansion model named STEAM, which leverages natural supervision in the existing taxonomy for expansion. To generate natural self-supervision signals, STEAM samples mini-paths from the existing taxonomy, and formulates a node attachment prediction task between anchor mini-paths and query terms. To solve the node attachment task, it learns feature representations for query-anchor pairs from multiple views and performs multi-view co-training for prediction. Extensive experiments show that STEAM outperforms state-of-the-art methods for taxonomy expansion by 11.6% in accuracy and 7.0% in mean reciprocal rank on three public benchmarks. The code and data for STEAM can be found at https://github.com/yueyu1030/STEAM.