Autonomous spacecraft relative navigation technology has been planned for and applied to many famous space missions. The development of on-board electronics systems has enabled the use of vision-based and LiDAR-based methods to achieve better performances. Meanwhile, deep learning has reached great success in different areas, especially in computer vision, which has also attracted the attention of space researchers. However, spacecraft navigation differs from ground tasks due to high reliability requirements but lack of large datasets. This survey aims to systematically investigate the current deep learning-based autonomous spacecraft relative navigation methods, focusing on concrete orbital applications such as spacecraft rendezvous and landing on small bodies or the Moon. The fundamental characteristics, primary motivations, and contributions of deep learning-based relative navigation algorithms are first summarised from three perspectives of spacecraft rendezvous, asteroid exploration, and terrain navigation. Furthermore, popular visual tracking benchmarks and their respective properties are compared and summarised. Finally, potential applications are discussed, along with expected impediments.