The Satellite network is an important part of the global network. However, the complex architecture, changeable constellation topology, and frequent inter-satellite connection switching problems bring great challenges to the routing designs of satellite networks, making the study of the routing methods in satellite networks a research hotspot. Therefore, this paper investigates the latest existing routing works to tackle the dynamic routing problems in satellite networks. The architecture and development of satellite networks are presented and analyzed first. Afterward, dynamic routing problems in satellite networks are analyzed in detail based on the time-varying network topology. According to the latest works, the advanced satellite network routing schemes, including single-layer and multi-layer dynamic routing are introduced and analyzed. In addition, the merits, shortcomings, and applications of these schemes are analyzed and summarized. Finally, potential technologies and future directions are discussed.