The talk will briefly introduce the EGI e-infrastructure and how its members, the NGIs, and partners are tacking the challenges of advanced computing to facilitate Open Science today, and tomorrow in EOSC. The EGI federation offers various types of storage, compute, data and application services, as well as related consultancy and training for structured research communities and individual researchers. These services are used by over 200 international communities and projects, of which more than 30 are featured on the ESFRI research infrastructure roadmap. Moreover, through recent H2020 initiatives, primarily EOSC-hub, the same services are used to establish the foundation of the European Open Science Cloud. The talk will highlight the recurring challenges that international communities face when using distributed computing and data services from EGI, as well as the technological and policy challenges and solutions that EOSC-hub brings to transition the fragmented e-infrastructure landscape to EOSC.