In the vision of Global Research Data Infrastructures (GRDIs), data storage and management plays a crucial role. A successful GRDI will require a common globally interoperable distributed data system, formed out of data centres, that incorporates emerging technologies and new scientific data activities. The main challenge is to define common certification and auditing frameworks that will allow storage providers and data communities to build a viable partnership based on trust. To achieve this, it is necessary to find a long-term commitment model that will give financial, legal, and organisational guarantees of digital information preservation. In this article we discuss the state of the art in data storage and management for GRDIs and point out future research directions that need to be tackled to implement GRDIs.