The scientific network cumulants, concentration and superconcentration focuses on the common interests of nine young scientists at the research locations Berlin, Bochum, Münster and Osnabrück. The three topics are fundamental in several research areas in current probability theory. The main focus of the first phase of the scientific network is on the one hand to explore new applications of the classical method of cumulants to prove concentration inequalities or moderate deviations. On the other hand the technical tools whose descriptions are scattered throughout the literature should be brought together in an survey article. We take this opportunity to review the optimality of the given assumptions and adapt them to new applications. Moreover, in a second phase we pick up the very young research topic of superconcentration and develop a deeper understanding of criteria accompanying the phenomenon of superconcentration. For this purpose we form smaller working groups and implement five focused workshops at the home institutions of the applicants. A group of distinguished international experts will accompany this programme and will give fresh input for further research.