Abstract: 6G communication systems would pose higher demands on latency and network response abilities, while the edge computing could reduce latency to improve response capabilities and reduce data traffic between users and centralized cloud computing resources. The satellite-terrestrial network is one of promising trends in the future 6G communication systems, and the embedding of edge computing into satellite-terrestrial networks turns out to be one important field. On the basis of existing works on edge computing in the satellite-terrestrial network, this survey first proposes the mobile edge computing-enabled network’s system architecture, summarizes the use cases, and logic functionality architecture, then concludes the existing computing offloading techniques, and finally presents the key challenges in the future.