Benefiting from maneuverability, flexibility, and low-cost deployment, aerial base stations (ABSs) have emerged as a promising solution to meet the coverage demand when terrestrial BSs are overloaded and unavailable. Nevertheless, the high mobility of ABSs and the complicated interference incurred by the addition of ABSs inevitably cause the spatial-temporal discontinuity in air-to-ground (A2G) coverage, which renders the network unable to provide users with on-demand coverage. On this account, this article discusses how to enhance the A2G coverage by exploiting the ever more enhanced computation capability of network edge nodes. In particular, we propose a coverage-oriented computing control architecture for adaptive coverage structure generation and resource orchestration based on the designed optimal deployment scheme for ABSs. This architecture can flexibly adjust the coverage structure and available resources to ensure the spatial continuity in A2G coverage. Furthermore, we design an efficient aerial-computing-based resource management scheme for ABSs to enable temporal continuity in A2G coverage by exploiting artificial intelligence approaches.