Abstract:SAR images possess unique attributes that present challenges for both human observers and vision AI models to interpret, owing to their electromagnetic characteristics. The interpretation of SAR images encounters various hurdles, with one of the primary obstacles being the data itself, which includes issues related to both the quantity and quality of the data. The challenges can be addressed using generative AI technologies. Generative AI, often known as GenAI, is a very advanced and powerful technology in the field of artificial intelligence that has gained significant attention. The advancement has created possibilities for the creation of texts, photorealistic pictures, videos, and material in various modalities. This paper aims to comprehensively investigate the intersection of GenAI and SAR. First, we illustrate the common data generation-based applications in SAR field and compare them with computer vision tasks, analyzing the similarity, difference, and general challenges of them. Then, an overview of the latest GenAI models is systematically reviewed, including various basic models and their variations targeting the general challenges. Additionally, the corresponding applications in SAR domain are also included. Specifically, we propose to summarize the physical model based simulation approaches for SAR, and analyze the hybrid modeling methods that combine the GenAI and interpretable models. The evaluation methods that have been or could be applied to SAR, are also explored. Finally, the potential challenges and future prospects are discussed. To our best knowledge, this survey is the first exhaustive examination of the interdiscipline of SAR and GenAI, encompassing a wide range of topics, including deep neural networks, physical models, computer vision, and SAR images. The resources of this survey are open-source at \url{https://github.com/XAI4SAR/GenAIxSAR}.