Space-air-ground integrated networks (SAGINs) have gained significant attention and become a promising architecture for ubiquitous connectivity for SG-Advanced and 6G, enabling the integration of satellite networks, aerial networks, and terrestrial networks. This integration brings tremendous communication benefits, such as non-terrestrial networks, seamless global coverage, high flexibility, and augmented system capacity. Meanwhile, computing capability becomes an indispensable part of the SAGIN ecosystem. In SAGINs, limited and unbalanced computation and communication resources of different network segments make it challenging to provide strict quality-of-service (QoS) guarantees for specific traffic (e.g., delay-sensitive traffic and outage-sensitive traffic). To fully utilize available system resources in SAGINs, cooperative computing among different network segments is a promising technology. This article presents the fundamentals and applications of computing over SAGINs by introducing the system architecture and explaining the potential technical issues related to cooperative computing. Furthermore, the potential for cooperative computing over SAGINs is showcased with a preliminary performance evaluation. Finally, future research opportunities are discussed.