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 phase transition in Fermi systems- A highly correlated Fermi liquid in heavy-fermion metals.- Metals with a strongly correlated electron liquid.- Transverse magnetoresistance in the HF metal CeCoIn5- Magnetic-field-induced reentrance of Fermi-liquid behavior and spin-lattice relaxation rates in YbCu5-xAux.- Relationships between critical magnetic fields Bc0 and Bc2 in HF metals and high-Tc superconductors.- Quantum criticality of heavy-fermion systems.- Scaling behavior at quantum criticality.- The impact of fermion-condensation quantum phase transition on ordinary continuous phase transitions in HF metals.- Asymmetric conductivity of strongly correlated metals.- Violation of the Wiedemann-Franz law in strongly correlated Fermi systems.- Conclusions.