PART I Normal Fermi-liquid Elementary excitations in quantum Fermi-liquid Quasi-particle interaction Effective mass Compressibility and sonic speed Zero sound Spin waves in Fermi-liquid Magnetic susceptibility Kinetic equation for quasi-particles of neutral Fermi-liquid Neutral and charged Fermi-systems Green functions for Fermi-system at zero temperature Green functions for macroscopic system Energy spectrum determination by means of Green function Green function for ideal Fermi-gas Fermi-particle momentum distribution Calculation of thermodynamical quantities by means of Green function Interaction representation Diagrammatic technique for Fermi-systems Properly energetic function Two-particle Green function Connection of vertex function with scattering amplitude and interaction function Identities for derivatives of Green functions Connection between maximum momentum and density Almost ideal Fermi-gas with repulsion between particles Green function for almost ideal Fermi-gas Superfluidity theory Elementary excitations in Bose-liquid Superfluidity criterion Energy spectrum of He-4 Basic experimental facts in superfluidity On the superfluiduty in Fermi- and Bose-systems Superfluid liquid mixtures: He-3 and He-4 Nuclear matter Matter in central areas of neutron stars Thermodynamical functions for superfluid He-4 Thermodynamical contribution of roton and phonon gases Whether do vertices make contribution into thermodynamics? Types of sonic excitations in He-4 The 1st and the 2nd sonic speed Possibility of its generation and observation Diagrammatic technique for Bose-liquid Properly energetic functions Green function behavior at small momenta Quasi-particle decay Analytical properties of Green functions Quantized vortex filaments Spectrum properties at the point of its finish PART II Green functions at finite temperatures Thermal Green functions Green function and diagrammatic technique application in superconductivity Superfluid Fermi-gas Energy spectrum, thermodynamical properties Green functions for superfluid Fermi-gas System of basic equations for the superconductor Phonon model of the superconductor Green functions in the theory of magnetic systems Kondo effect Green functions for radiation in the absorbing medium Dielectric coefficient calculation Electromagnetic wave absorption in a plasma Interaction of electrons with lattice oscillations in normal metals Peculiarities of complex metal systems Electron liquid in dimensionally quantized layers Particle passing through a plasma Molecular interaction between solid bodies Green functions in nucleon-nuclear systems Peculiarities of its application for superfluidity description in atomic nuclei Problems of correctness of such a description Review: model calculations of the energy gap in superfluid nucleon systems Probable displays of superfluidity in dynamics and evolution of neutron stars Approximation problems of Green functions and observed properties of physical systems using Padet-approximation   LITERATURE: A.A. Abrikosov, L.P. Gor’kov, I.E. Dzyaloshinsky. Quantum field theory methods in statistical physics E.M. Lifshitz, L.P Pitaevsky. Statistical physics (vol.9) D.A. Kirzhnits. Field methods in many particle theory A.S. Kondrat’ev, A.E. Kuchma. Lectures on quantum liquids G. Shriffer. Theory of superconductivity