Bioimage data are generated in diverse research fields throughout the life and biomedical sciences. Its potential for advancing scientific progress via modern, data‐driven discovery approaches reaches beyond disciplinary borders. To fully exploit this potential, it is necessary to make bioimaging data, in general, multidimensional microscopy images and image series, FAIR, that is, findable, accessible, interoperable and reusable. These FAIR principles for research data management are now widely accepted in the scientific community and have been adopted by funding agencies, policymakers and publishers. To remain competitive and at the forefront of research, implementing the FAIR principles into daily routines is an essential but challenging task for researchers and research infrastructures. Imaging core facilities, well‐established providers of access to imaging equipment and expertise, are in an excellent position to lead this transformation in bioimaging research data management. They are positioned at the intersection of research groups, IT infrastructure providers, the institution´s administration, and microscope vendors. In the frame of German BioImaging – Society for Microscopy and Image Analysis (GerBI‐GMB), cross‐institutional working groups and third‐party funded projects were initiated in recent years to advance the bioimaging community's capability and capacity for FAIR bioimage data management. Here, we provide an imaging‐core‐facility‐centric perspective outlining the experience and current strategies in Germany to facilitate the practical adoption of the FAIR principles closely aligned with the international bioimaging community. We highlight which tools and services are ready to be implemented and what the future directions for FAIR bioimage data have to offer.