Recent years have witnessed a significant advancement in brain imaging techniques that offer a non-invasive approach to mapping the structure and function of the brain. Concurrently, generative artificial intelligence (AI) has experienced substantial growth, involving using existing data to create new content with a similar underlying pattern to real-world data. The integration of these two domains, generative AI in neuroimaging, presents a promising avenue for exploring various fields of brain imaging and brain network computing, particularly in the areas of extracting spatiotemporal brain features and reconstructing the topological connectivity of brain networks. Therefore, this study reviewed the advanced models, tasks, challenges, and prospects of brain imaging and brain network computing techniques and intends to provide a comprehensive picture of current generative AI techniques in brain imaging. This review is focused on novel methodological approaches and applications of related new methods. It discussed fundamental theories and algorithms of four classic generative models and provided a systematic survey and categorization of tasks, including co-registration, super-resolution, enhancement, classification, segmentation, cross-modality, brain network analysis, and brain decoding. This paper also highlighted the challenges and future directions of the latest work with the expectation that future research can be beneficial.