By integrating UAVs into terrestrial communication networks, air-ground integrated networks (AGIN) have shown great potential to revolutionize wireless communications and enhance ubiquitous wireless services. However, limited spectrum resource has been identified as a bottleneck to release the benefits of AGIN. Spectrum sharing is a powerful technique to improve spectrum utilization and poses new opportunities and challenges in AGIN. This article first elaborates the characteristics in air-ground spectrum sharing from state uncertainties and security threats perspectives. Then, a robust spectrum sharing framework is developed in which spectrum utilizing networks, spectrum monitoring networks, and spectrum clouds cooperatively form a cognitive dynamic close-loop control system. Furthermore, we discuss the key enabling technologies for robust spectrum sharing in AGIN to tackle the issues of uncertain and unknown information. Next, an exemplary robust spectrum sharing case with location uncertainty is provided. Finally, the technical challenges and open issues ahead are presented.