Data stewardship is generally understood as encompassing the collection, digitasation, maintenance, curation, storage, analysis, sharing, use and reuse of digital datasets. Since digitation processes begun in the GLAM (Galleries, Libraries, Archives, Museums) sector many issues and concerns have arisen in the context of digital datasets belonging to the Indigenous and Minorities Communities and the rights of Indigenous peoples in decision-making concerning their digital cultural heritage. Indigenous data governance includes both the stewardship and the processes necessary to implement Indigenous control over Indigenous data. However, legislation and practices endorsing open data and open science overlooks the rights and interests of Indigenous and Minorities Communities, failing to adequately consider and respect their rights and interests. This paper examines, from an interdisciplinarity perspective, how to develop an open data stewardship policy and protocols for the governance of the indigenous and minority communities’ cultural heritage datasets within the GLAM environment. It focuses particularly on the implementation and application of both FAIR (Findable, Accessible, Interoperable, Reusable) and CARE (Collective Benefit, Authority to Control, Responsibility, Ethics) principles in the GLAM sector. Furthermore, this paper reflects indeed on the issues and challenges posed for data governance and stewardship by the adoption, at EU level, of the Open Data Directive and the Artificial Intelligence Act. The objective is to shed light on how to develop a sustainable, resilient, and accountable data stewardship strategy coherent and consistent with the right of indigenous people and minority communities to foster a cultural governance and sovereignty over their data.