Relational tables on the Web store a vast amount of knowledge. Owing to the wealth of such tables, there has been tremendous progress on a variety of tasks in the area of table understanding. However, existing work generally relies on heavily-engineered task-specific features and model architectures. In this paper, we present TURL, a novel framework that introduces the pre-training/fine-tuning paradigm to relational Web tables. During pre-training, our framework learns deep contextualized representations on relational tables in a self-supervised manner. Its universal model design with pre-trained representations can be applied to a wide range of tasks with minimal task-specific fine-tuning. Specifically, we propose a structure-aware Transformer encoder to model the row-column structure, and present a new Masked Entity Recovery (MER) objective for pre-training to capture relational knowledge. We compiled a benchmark consisting of 6 different tasks for table understanding and used it to systematically evaluate TURL. We show that TURL generalizes well to all tasks and substantially outperforms existing methods in almost all instances.