Edge-computing-enhanced Internet of Vehicles (EC-IoV) enables ubiquitous data processing and content sharing among vehicles and terrestrial edge computing (TEC) infrastructures (e.g., 5G base stations and roadside units) with little or no human intervention, and plays a key role in the intelligent transportation systems. However, EC-IoV is heavily dependent on the connections and interactions between vehicles and TEC infrastructures, thus will break down in some remote areas where TEC infrastructures are unavailable (e.g., desert, isolated islands, and disaster-stricken areas). Driven by the ubiquitous connections and global-area coverage, space–air–ground-integrated networks (SAGINs) efficiently support seamless coverage and efficient resource management, and represent the next frontier for edge computing. In light of this, we first review the state-of-the-art edge computing research for SAGINs in this article. After discussing several existing orbital and aerial edge computing architectures, we propose a framework of edge computing-enabled SAGINs to support various Internet of Vehicles (EC-IoV) services for the vehicles in remote areas. The main objective of the framework is to minimize the task completion time and satellite resource usage. To this end, a preclassification scheme is presented to reduce the size of action space, and a deep imitation learning-driven offloading and caching algorithm is proposed to achieve real-time decision making. The simulation results show the effectiveness of our proposed scheme. Finally, we also discuss some technology challenges and future directions.