This review discusses the instabilities of the Fermi-liquid state of conduction electrons in metals, focusing on magnetic quantum phase transitions. It covers both theoretical concepts and experimental data on selected materials to assess the validity of current theories. The review begins with a brief overview of Fermi-liquid theory, including the Kondo effect in local-moment and Kondo-lattice systems. It then delves into the theoretical approach to quantum phase transitions in metallic systems, particularly the Hertz-Millia-Moriya theory, and its limitations. The experimental section examines heavy-fermion materials and transition-metal alloys, comparing their behavior with existing theories. The review highlights the challenges in distinguishing between non-Fermi-liquid behavior and the breakdown of the Fermi-liquid concept, and discusses the conditions under which power-law behavior is expected. It also explores the requirements for observing non-Fermi-liquid exponents and the role of scaling in understanding critical behavior. The review concludes with a detailed analysis of the Kondo effect and the concept of a local Fermi liquid, including the Anderson and Kondo lattice models, and their implications for the phase diagram of these systems.This review discusses the instabilities of the Fermi-liquid state of conduction electrons in metals, focusing on magnetic quantum phase transitions. It covers both theoretical concepts and experimental data on selected materials to assess the validity of current theories. The review begins with a brief overview of Fermi-liquid theory, including the Kondo effect in local-moment and Kondo-lattice systems. It then delves into the theoretical approach to quantum phase transitions in metallic systems, particularly the Hertz-Millia-Moriya theory, and its limitations. The experimental section examines heavy-fermion materials and transition-metal alloys, comparing their behavior with existing theories. The review highlights the challenges in distinguishing between non-Fermi-liquid behavior and the breakdown of the Fermi-liquid concept, and discusses the conditions under which power-law behavior is expected. It also explores the requirements for observing non-Fermi-liquid exponents and the role of scaling in understanding critical behavior. The review concludes with a detailed analysis of the Kondo effect and the concept of a local Fermi liquid, including the Anderson and Kondo lattice models, and their implications for the phase diagram of these systems.