Reference anatomies of the brain ('templates') and corresponding atlases are the foundation for reporting standardized neuroimaging results. Currently, there is no registry of templates and atlases; therefore, the redistribution of these resources occurs either bundled within existing software or in ad hoc ways such as downloads from institutional sites and general-purpose data repositories. We introduce TemplateFlow as a publicly available framework for human and non-human brain models. The framework combines an open database with software for access, management, and vetting, allowing scientists to share their resources under FAIR-findable, accessible, interoperable, and reusable-principles. TemplateFlow enables multifaceted insights into brains across species, and supports multiverse analyses testing whether results generalize across standard references, scales, and in the long term, species.