Introduction.- Landau - Fermi liquid theory.- Fermi liquid with fermion condensation.- The topological phase transitions related to fermion condensate.- Appearance of fermion-condensation quantum phase transition in Fermi systems.- The superconducting state with a fermion condensate.- The dispersion law and lineshape of single-particle excitations.- Electron liquid with fermion condensate in magnetic fields.- Emergence of fermion-condensation quantum...