Ambient intelligence (AmI) is a computing paradigm in which physical environments are embedded with sensing, computation, and communication so they can perceive people and context, decide appropriate actions, and respond autonomously. Realizing AmI at global scale requires sixth generation (6G) wireless networks with capabilities for real time perception, reasoning, and action aligned with human behavior and mobility patterns. We argue that Generative Artificial Intelligence (GenAI) is the creative core of such environments. Unlike traditional AI, GenAI learns data distributions and can generate realistic samples, making it well suited to close key AmI gaps, including generating synthetic sensor and channel data in under observed areas, translating user intent into compact, semantic messages, predicting future network conditions for proactive control, and updating digital twins without compromising privacy. This chapter reviews foundational GenAI models, GANs, VAEs, diffusion models, and generative transformers, and connects them to practical AmI use cases, including spectrum sharing, ultra reliable low latency communication, intelligent security, and context aware digital twins. We also examine how 6G enablers, such as edge and fog computing, IoT device swarms, intelligent reflecting surfaces (IRS), and non terrestrial networks, can host or accelerate distributed GenAI. Finally, we outline open challenges in energy efficient on device training, trustworthy synthetic data, federated generative learning, and AmI specific standardization. We show that GenAI is not a peripheral addition, but a foundational element for transforming 6G from a faster network into an ambient intelligent ecosystem.