Research Infrastructures (RI) and Core Facilities (CF) are strong drivers for generating research results and, thus, knowledge at European Universities. In this paper, we provide insight into different features of RI and CF, their organisational structure and governance, funding mechanisms, critical factors for success, and challenges and opportunities associated with implementing and operating these research support structures. Our results are based on a comparative analysis across six European universities from the 4EU + University Alliance. Due to the lack of a clear definition of RI and CF, we provide a variety of indicators and criteria attributed to such facilities and highlight differences between CF and RI in terms of goals and objectives. If establishing Core Facilities is seen as a response to the evolving needs and challenges of researchers, a centralised management and collaborative governance model can provide a practical solution. We explore the legal framework, organisation, access, users and charge rate models at Core Facilities. We also identify several challenges in setting up and maintaining Core Facilities. Special attention is directed towards staff challenges, including introducing the staff category of 'scientific technical experts'. Finally, we present a Core Facility Manifesto and share our conclusions.