Recent years have witnessed the burgeoning of pretrained language models (LMs) for text-based natural language (NL) understanding tasks. Such models are typically trained on free-form NL text, hence may not be suitable for tasks like semantic parsing over structured data, which require reasoning over both free-form NL questions and structured tabular data (e.g., database tables). In this paper we present TABERT, a pretrained LM that jointly learns representations for NL sentences and (semi-)structured tables. TABERT is trained on a large corpus of 26 million tables and their English contexts. In experiments, neural semantic parsers using TABERT as feature representation layers achieve new best results on the challenging weakly-supervised semantic parsing benchmark WIKITABLEQUESTIONS, while performing competitively on the text-toSQL dataset SPIDER.