Text-to-SQL parsing is an essential and challenging task. The goal of text-to-SQL parsing is to convert a natural language (NL) question to its corresponding structured query language (SQL) based on the evidences provided by relational databases. Early text-to-SQL parsing systems from the database community achieved a noticeable progress with the cost of heavy human engineering and user interactions with the systems. In recent years, deep neural networks have significantly advanced this task by neural generation models, which automatically learn a mapping function from an input NL question to an output SQL query. Subsequently, the large pre-trained language models have taken the state-of-the-art of the text-to-SQL parsing task to a new level. In this survey, we present a comprehensive review on deep learning approaches for text-to-SQL parsing. First, we introduce the text-to-SQL parsing corpora which can be categorized as single-turn and multi-turn. Second, we provide a systematical overview of pre-trained language models and existing methods for text-to-SQL parsing. Third, we present readers with the challenges faced by text-to-SQL parsing and explore some potential future directions in this field.