  We study higher moments of convolutions of the characteristic function of a
set, which generalize a classical notion of the additive energy. Such
quantities appear in many problems of additive combinatorics as well as in
number theory. In our investigation we use different approaches including basic
combinatorics, Fourier analysis and eigenvalues method to establish basic
properties of higher energies. We provide also a sequence of applications of
higher energies additive combinatorics.
