The COVID-19 pandemic has had a significant impact on society, both because of the serious health effects of COVID-19 and because of public health measures implemented to slow its spread. Many of these difficulties are fundamentally information needs; attempts to address these needs have caused an information overload for both researchers and the public. Natural language processing (NLP), the branch of artificial intelligence that interprets human language, can be applied to address many of the information needs made urgent by the COVID-19 pandemic. This review surveys approximately 150 NLP studies and more than 50 systems and datasets addressing the COVID-19 pandemic. We detail work on four core NLP tasks: information retrieval, named entity recognition, literature-based discovery, and question answering. We also describe work that directly addresses aspects of the pandemic through four additional tasks: topic modeling, sentiment and emotion analysis, caseload forecasting, and misinformation detection. We conclude by discussing observable trends and remaining challenges.