We present Covidex, a search engine that exploits the latest neural ranking models to provide information access to the COVID-19 Open Research Dataset curated by the Allen Institute for AI. Our system has been online and serving users since late March 2020. The Covidex is the user application component of our three-pronged strategy to develop technologies for helping domain experts tackle the ongoing global pandemic. In addition, we provide robust and easy-to-use keyword search infrastructure that exploits mature fusion-based methods as well as standalone neural ranking models that can be incorporated into other applications. These techniques have been evaluated in the multi-round TREC-COVID challenge: Our infrastructure and baselines have been adopted by many participants, including some of the best systems. In round 3, we submitted the highest-scoring run that took advantage of previous training data and the second-highest fully automatic run. In rounds 4 and 5, we submitted the highest-scoring fully automatic runs.