Unmanned aerial vehicle (UAV)-enabled communication networks are promising in the fifth and beyond wireless communication systems. In this paper, we shed light on three UAV-enabled mobile edge computing (MEC) architectures. Those architectures have been receiving ever increasing research attention for improving computation performance and decreasing execution latency by integrating UAV into MEC networks. We present a comprehensive survey for the state-of-the-art research in this domain. Important implementation issues are clarified. Moreover, in order to provide an enlightening guidance for future research directions, key challenges and open issues are discussed.