Research Centres hosting researchers and organizing scientific events at a high level. Three different types of centres can be distinguished according to their aim: a) centres organizing high level one week conferences in mathematics or their interface with sciences and industry; b) centres organizing, over three or more months, targeted advanced scientific programmes at doctoral level or on specific research challenges; c) high-level research institutes with few permanent positions and a highly developed visitor’s programme. (High level mathematics, interface between mathematics and other sciences).