The widespread popularity of unmanned aerial vehicles enables an immense amount of power lines inspection data to be collected. How to employ massive inspection data especially the visible images to maintain the reliability, safety, and sustainability of power transmission is a pressing issue. To date, substantial works have been conducted on the analysis of power lines inspection data. With the aim of providing a comprehensive overview for researchers who are interested in developing a deep-learning-based analysis system for power lines inspection data, this paper conducts a thorough review of the current literature and identifies the challenges for future research. Following the typical procedure of inspection data analysis, we categorize current works in this area into component detection and fault diagnosis. For each aspect, the techniques and methodologies adopted in the literature are summarized. Some valuable information is also included such as data description and method performance. Further, an in-depth discussion of existing deep-learning-related analysis methods in power lines inspection is proposed. Finally, we conclude the paper with several research trends for the future of this area, such as data quality problems, small object detection, embedded application, and evaluation baseline.