Tables on the Web contain a vast amount of knowledge in a structured form. To tap into this valuable resource, we address the problem of table retrieval: answering an information need with a ranked list of tables. We investigate this problem in two different variants, based on how the information need is expressed: as a keyword query or as an existing table ("query-by-table"). The main novel contribution of this work is a semantic table retrieval framework for matching information needs (keyword or table queries) against tables. Specifically, we (i) represent queries and tables in multiple semantic spaces (both discrete sparse and continuous dense vector representations) and (ii) introduce various similarity measures for matching those semantic representations. We consider all possible combinations of semantic representations and similarity measures and use these as features in a supervised learning model. Using two purpose-built test collections based on Wikipedia tables, we demonstrate significant and substantial improvements over state-of-the-art baselines.