1 Introduction 1.1 General considerations 1.2 Strong and weak interparticle interactions 1.3 Theoretical approaches to strongly correlated systems 1.4 Quantum phase transitions and NFL behavior of HF compounds 1.5 Main goals of the book References 2 Landau Fermi liquid theory 2.1 Quasiparticle paradigm 2.2 Pomeranchuk stability conditions 2.3 Thermodynamic and transport properties 2.3.1 Equation for the effective mass References 3 Density Functional Theory of Fermion Condensation 3.1 Introduction 3.2 Functional equation for the effective interaction 3.3 DFT and fermion condensation 3.4 DFT, the fermion condensation and superconductivity 3.5 Summary References 4 Topological fermion condensation quantum phase transition 4.1 The fermion-condensation quantum phase transition 4.1.1 The FCQPT order parameter 4.1.2 Quantum protectorate related to FCQPT 4.1.3 The in uence of FCQPT at nite temperatures 4.1.4 Two Scenarios of the Quantum Critical Point 4.1.5 Phase diagram of Fermi system with FCQPT 4.2 Topological phase transitions related to FCQPT References 5 Rearrangement of the single particle degrees of freedom 5.1 Introduction 5.2 Basic properties of systems with the FC 5.2.1 The case Tc 5.2.2 The case T 5.3 Validity of the quasiparticle pattern 5.3.1 Finite systems 5.3.2 Macroscopic systems 5.4 Interplay between fermion condensation and density-wave instability 5.5 Discussion References 6 Topological FCQPT in strongly correlated Fermi systems 6.1 The superconducting state with FC at T = 0 6.1.1 Green's function of the superconducting state with FC at T = 0 6.1.2 The superconducting state at nite temperatures 6.1.3 Bogolyubov quasiparticles 6.1.4 The dependence of superconducting phase transition temperature Tc on doping 6.1.5 The gap and heat capacity near Tc 6.2 The dispersion law and lineshape of single-particle excitations 6.3 Electron liquid with FC in magnetic elds 6.3.1 Phase diagram of electron liquid in magnetic eld 6.3.2 Magnetic eld dependence of the effective mass in HF metals and high-Tc superconductors 6.4 Appearance of FCQPT in HF compounds References 7 Effective mass and its scaling behavior 7.1 Scaling behavior of the effective mass near the topological FCQPT 7.2 T/B scaling in heavy fermion compounds References 8 Quantum spin liquid in geometrically frustrated magnets and the new state of matter 8.1 Introduction 8.2 Fermion condensation 8.3 Scaling of the physical properties 8.4 The frustrated insulator Herbertsmithite ZnCu3(OH)6Cl2 8.4.1 Thermodynamic properties References 9 One dimensional quantum spin liquid 9.1 Introduction 9.2 General considerations 9.3 Scaling of the thermodynamic properties 9.4 T - H phase diagram of 1D spin liquid 9.5 Discussion and summary References 10 Dynamic magnetic susceptibility of quantum spin liquid 10.1 Dynamic spin susceptibility of quantum spin liquids and HF metals 10.2 Theory of dynamic spin susceptibility of quantum spin liquid and heavy-fermion metals 10.3 Scaling behavior of the dynamic susceptibility References 11 Spin-lattice relaxation rate and optical conductivity of quantum spin liquid 11.1