Selecting patients for clinical trials is very labor-intensive. Our goal is to develop an automated system that can support doctors in this task. This paper describes a major step towards such a system: the automatic translation of clinical trial eligibility criteria from natural language into formal, logic-based queries. First, we develop a semantic annotation process that can capture many types of clinical trial criteria. Then, we map the annotated criteria to the formal query language. We have built a prototype system based on state-of-the-art NLP tools such as Word2Vec, Stanford NLP tools, and the MetaMap Tagger, and have evaluated the quality of the produced queries on a number of criteria from clinicaltrials.gov. Finally, we discuss some criteria that were hard to translate, and give suggestions for how to formulate eligibility criteria to make them easier to translate automatically.