In recent years, generative artificial intelligence is progressively introduced into the field of radio spectrum cognition due to its powerful capabilities in data distribution fitting, data generation, and data completion. Compared to conventional approaches rely on physical modeling, mathematical interpolation, and discriminative artificial intelligence techniques, generative AI has significantly enhanced the accuracy of radio spectrum cognition. This paper systematically reviewed the research progress of generative artificial intelligence in radio spectrum cognition, with a focused analysis on the technical principles, application scenarios, and representative works of different generative paradigms. The challenges faced by generative AI in spectrum cognition were further discussed, including scarce training data, limited generalization in unknown scenarios, and insufficient model interpretability. In the future, by cross-modal knowledge fusion, physics-informed embedding, and the establishment of a trustworthy assessment framework, generative artificial intelligence is expected to advance radio spectrum cognition toward high precision, robust generalization, and enhanced interpretability, thereby effectively supporting the efficient utilization of spectrum resources.