In recent years, the increasing complexity of space missions has led to an exponential growth in space-generated data. However, limited satellites-to-ground bandwidth and scarce frequency resources pose significant challenges to traditional bent-pipe architecture, which faces severe transmission bottlenecks. In addition, onboard data must wait for satellites to pass over ground stations before transmission. The large-scale construction of ground stations is not only cost-prohibitive but also carries geopolitical and economic risks. Satellite edge computing has emerged as a promising solution to these bottlenecks by integrating mobile edge computing technology into satellite edges. This approach significantly enhances user experience and reduces redundant network traffic. By enabling onboard data processing, satellite edge computing shortens data acquisition times and reduces reliance on extensive ground station infrastructure. Furthermore, the integration of artificial intelligence (AI) and edge computing technologies offers an efficient and forward-looking path to address existing challenges. This study reviews the latest progress in intelligent satellite edge computing. First, the demands and applications of satellite edge computing in various typical scenarios are discussed. Next, key challenges and recent research advancements in this field are analyzed. Finally, several open research topics are highlighted, and new ideas are proposed to guide future studies. This discussion aims to provide valuable insights to promote technological innovation and the practical implementation of satellite edge computing.