With the continuous advancement of network technology, various emerging complex networking optimization problems have created a wide range of applications utilizing game theory. However, since game theory is a mathematical framework, game theory-based solutions often rely heavily on the experience and knowledge of human experts. Recently, the remarkable advantages exhibited by generative artificial intelligence (GAI) have gained widespread attention. In this work, we propose a novel GAI-enabled game theory solution that combines the powerful reasoning and generation capabilities of GAI to the design and optimization of mobile networking. Specifically, we first outline the game theory and key technologies of GAI, and explore the advantages of combining GAI with game theory. Then, we review the contributions and limitations of existing research and demonstrate the potential application values of GAI applied to game theory in mobile networking. Subsequently, we develop a large language model (LLM)-enabled game theory framework to realize this combination, and demonstrate the effectiveness of the proposed framework through a case study in secured UAV networks. Finally, we provide several directions for future extensions.