Foundation models (FMs), large neural networks pretrained on extensive and diverse datasets, have revolutionized artificial intelligence and shown significant promise in medical imaging by enabling robust performance with limited labeled data. Although numerous surveys have reviewed the application of FM in healthcare care, brain imaging remains underrepresented, despite its critical role in the diagnosis and treatment of neurological diseases using modalities such as MRI, CT, and PET. Existing reviews either marginalize brain imaging or lack depth on the unique challenges and requirements of FM in this domain, such as multimodal data integration, support for diverse clinical tasks, and handling of heterogeneous, fragmented datasets. To address this gap, we present the first comprehensive and curated review of FMs for brain imaging. We systematically analyze 161 brain imaging datasets and 86 FM architectures, providing information on key design choices, training paradigms, and optimizations driving recent advances. Our review highlights the leading models for various brain imaging tasks, summarizes their innovations, and critically examines current limitations and blind spots in the literature. We conclude by outlining future research directions to advance FM applications in brain imaging, with the aim of fostering progress in both clinical and research settings.