The rapid advancement of generative models has empowered modern AI systems to comprehend and produce highly sophisticated content, even achieving human-level performance in specific domains. However, these models are fundamentally constrained by \emph{catastrophic forgetting}, \ie~a persistent challenge where models experience performance degradation on previously learned tasks when adapting to new tasks. To address this practical limitation, numerous approaches have been proposed to enhance the adaptability and scalability of generative AI in real-world applications. In this work, we present a comprehensive survey of continual learning methods for mainstream generative AI models, encompassing large language models, multimodal large language models, vision-language-action models, and diffusion models. Drawing inspiration from the memory mechanisms of the human brain, we systematically categorize these approaches into three paradigms: architecture-based, regularization-based, and replay-based methods, while elucidating their underlying methodologies and motivations. We further analyze continual learning setups for different generative models, including training objectives, benchmarks, and core backbones, thereby providing deeper insights into the field. The project page of this paper is available at https://github.com/Ghy0501/Awesome-Continual-Learning-in-Generative-Models.