Cervical cancer is one of the most common cancers in daily life. Early detection and diagnosis can effectively help facilitate subsequent clinical treatment and management. With the growing advancement of artificial intelligence (AI) and deep learning (DL) techniques, an increasing number of computer-aided diagnosis (CAD) methods based on deep learning have been applied in cervical cytology screening. In this paper, we survey more than 70 publications since 2016 to provide a systematic and comprehensive review of DL-based cervical cytology screening. First, we provide a concise summary of the medical and biological knowledge pertaining to cervical cytology, since we hold a firm belief that a comprehensive biomedical understanding can significantly contribute to the development of CAD systems. Then, we collect a wide range of public cervical cytology datasets. Besides, image analysis approaches and applications including cervical cell identification, abnormal cell or area detection, cell region segmentation and cervical whole slide image diagnosis are summarized. Finally, we discuss the present obstacles and promising directions for future researches in automated cervical cytology screening.