Unmanned aerial vehicles (UAV) have been widely used in various fields because of their high mobility and portability. At the same time, due to the rapid development of artificial intelligence, people’s demand for computing is increasing, and the computing power of existing mobile computing devices cannot fully meet the users’ needs for network quality. Therefore, people have proposed mobile edge computing technology (MEC), but MEC still has some shortcomings, such as poor dynamic performance. Therefore, combining the two will have better results. Due to the limited battery capacity of UAVs, the continuity of the UAV Communication Network is affected. Therefore, effectively using the limited spectrum resources, reducing the energy consumption of UAVs and meeting the users’ quality of service needs have become urgent problems to be solved. Under the framework of UAV-assisted MEC, this paper focuses on the downlink communication energy efficiency of a single UAV Communication Network, arranges and analyzes the methods and technologies found in different research, reviews specific UAV practical applications, provides a relevant discussion and method analysis for the problems existing in the UAV Communication Network, and presents its existing problems and our future research direction.