Recent research has investigated quantum NLP, designing algorithms that process natural language in quantum computers, and also quantum-inspired algorithms that improve NLP performance on classical computers. In this survey, we review representative methods at the intersection of NLP and quantum physics in the past ten years, categorizing them according to the use of quantum theory, the linguistic targets that are modeled, and the downstream application. The literature review ends with a discussion on the key factors to the success that has been achieved by existing work, as well as challenges ahead, with the goal of better understanding the promises and further directions.