Due to the limitations of LiDAR, such as its high cost, short service life and massive volume, visual sensors with their lightweight and low cost are attracting more and more attention and becoming a research hotspot. As the hardware computation power and deep learning develop by leaps and bounds, new methods and ideas for dealing with visual simultaneous localization and mapping (VSLAM) problems have emerged. This paper systematically reviews the VSLAM methods based on deep learning. We briefly review the development process of VSLAM and introduce its fundamental principles and framework. Then, we focus on the integration of deep learning and VSLAM from three aspects: visual odometry (VO), loop closure detection, and mapping. We summarize and analyze the contribution and weakness of each algorithm in detail. In addition, we also provide a summary of widely used datasets and evaluation metrics. Finally, we discuss the open problems and future directions of combining VSLAM with deep learning.