  The Green-function technique, termed the irreducible Green functions (IGF)
method, that is a certain reformulation of the equation-of motion method for
double-time temperature dependent Green functions is presented. This method was
developed to overcome some ambiguities in terminating the hierarchy of the
equations of motion of double-time Green functions and to give a workable
technique to systematic way of decoupling. The approach provides a practical
method for description of the many-body quasi-particle dynamics of correlated
systems on a lattice with complex spectra. Moreover, it provides a very compact
and self-consistent way of taking into account the damping effects and finite
lifetimes of quasi-particles due to inelastic collisions. In addition, it
correctly defines the Generalized Mean Fields, that determine elastic
scattering renormalizations and, in general, are not functionals of the mean
particle densities only. Although some space is devoted to the formal structure
of the method, the emphasis is on its utility. Applications to the lattice
fermion models such as Hubbard/Anderson models and to the Heisenberg ferro- and
antiferromagnet, which manifest the operational ability of the method are
given. It is shown that the IGF method provides a powerful tool for the
construction of essentially new dynamical solutions for strongly interacting
many-particle systems with complex spectra.
