In existing computing systems, such as edge computing and cloud computing, several emerging applications and practical scenarios are mostly unavailable or only partially implemented. To overcome the limitations that restrict such applications, the development of a comprehensive computing paradigm has garnered attention in both academia and industry. However, a gap exists in the literature owing to the scarce research, and a comprehensive computing paradigm is yet to be systematically designed and reviewed. This study introduces a novel concept, called aerial computing, via the amalgamation of aerial radio access networks and edge computing, which attempts to bridge the gap. Specifically, first, we propose a novel comprehensive computing architecture that is composed of low-altitude computing, high-altitude computing, and satellite computing platforms, along with conventional computing systems. We determine that aerial computing offers several desirable attributes: global computing service, better mobility, higher scalability and availability, and simultaneity. Second, we comprehensively discuss key technologies that facilitate aerial computing, including energy refilling, edge computing, network softwarization, frequency spectrum, multiaccess techniques, artificial intelligence, and big data. In addition, we discuss vertical domain applications (e.g., smart cities, smart vehicles, smart factories, and smart grids) supported by aerial computing. Finally, we highlight several challenges that need to be addressed and their possible solutions.