This survey provides a comprehensive review on recent advancements of generative learning models in robotic manipulation, addressing key challenges in the field. Robotic manipulation faces critical bottlenecks, including significant challenges in insufficient data and inefficient data acquisition, long-horizon and complex task planning, and the multi-modality reasoning ability for robust policy learning performance across diverse environments. To tackle these challenges, this survey introduces several generative model paradigms, including Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), diffusion models, probabilistic flow models, and autoregressive models, highlighting their strengths and limitations. The applications of these models are categorized into three hierarchical layers: the Foundation Layer, focusing on data generation and reward generation; the Intermediate Layer, covering language, code, visual, and state generation; and the Policy Layer, emphasizing grasp generation and trajectory generation. Each layer is explored in detail, along with notable works that have advanced the state of the art. Finally, the survey outlines future research directions and challenges, emphasizing the need for improved efficiency in data utilization, better handling of long-horizon tasks, and enhanced generalization across diverse robotic scenarios. All the related resources, including research papers, open-source data, and projects, are collected for the community in https://github.com/GAI4Manipulation/AwesomeGAIManipulation