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Driven by the visions of Internet of Things (IoT), there is an ever-increasing demand for computation resources of IoT users to support diverse applications. Mobile edge computing (MEC) has been deemed a promising solution to settle the conflict between the resource-hungry mobile applications and the resource-constrained IoT users. On the other hand, in order to provide ubiquitous and reliable connectivity in wireless networks, unmanned aerial vehicles (UAVs) can be leveraged as efficient aerial platforms by exploiting their inherent attributes, such as the on-demand deployment, high cruising altitude, and controllable maneuverability in three-dimensional (3D) space. Thus, the UAV-enabled aerial MEC is believed as a win-win solution to facilitate cost-effective and energy-saving communication and computation services in various environments. In this paper, we provide a comprehensive survey on the UAV-enabled aerial MEC. Firstly, the related advantages and research challenges for aerial MEC are discussed. Then, we provide a comprehensive review of the recent research advances, which is categorized by different domains, including the joint optimization of UAV trajectory, computation offloading and resource allocation, UAV deployment, task scheduling and load balancing, interplay between aerial MEC and other technologies, as well as the machine-learning (ML)-driven optimization. Finally, some important research directions deserved more efforts in future work are summarized.