The Istituto Nazionale di Alta Matematica (acronym: INdAM) is a research institution created by the Italian law.The INdAM-COFUND-2012 Programme (Programme) plans to continue the previous INdAM-COFUND programme, supported under the Call FP7-PEOPLE-COFUND-2008. The Programme improves the existing Cofund programme by increasing the number of fellowships, increasing the salaries of the fellows, better balancing the number of incoming versus outgoing fellowships, increasing the total INdAM budget intended for fellowships, increasing the number of non-nationals members of the Panels and introducing an international supervisors Committee.The Programme, intended for experienced researchers in mathematics and/or its applications, aims to reinforce the international dimension of the career of Italian (national or assimilated) researchers, to reinforce the scientific excellence of Italy inside the ERA through knowledge sharing with incoming top-class researchers, to reinforce the attractiveness of Italy by encouraging reintegration in Italy of Italian researchers who have carried out research in a Third Country for at least 3 year.The fellowships to be awarded are of three types(1) Outgoing fellowships (with return phase)(2) Incoming fellowships(3) Re-integration fellowships.The Programme is open to all the researchers. Italian nationals (or assimilated) can apply for outgoing and reintegration fellowships, non-nationals for incoming fellowships.As to the choice of the research programs, the expression “mathematics and/or its applications” has a very broad meaning. It includes all the fields of mathematics, applications of mathematics to the natural and social sciences, e.g. physics, biology, chemistry, finance and economics, to computer science, to engineering, to industry and technology and so on.The duration of the Programme is 60 months. The total number of fellowships: 30. The total number of fellow-year: 60.