This review discusses the breakdown of the Fermi-liquid state of conduction electrons in metals near magnetic quantum phase transitions. It presents existing theoretical concepts and experimental data on selected materials to assess the validity of current theory. After recalling Fermi-liquid theory, the review describes the local Fermi-liquid state in quantum impurity models and their lattice versions. Scaling concepts applicable to quantum phase transitions are then presented, with a detailed description of the Hertz-Millis-Moriya theory. The breakdown of this theory is analyzed in several examples. Experimental data on heavy-fermion materials and transition-metal alloys are reviewed and compared with existing theory. The review covers the fundamentals of Fermi-liquid theory, including the quasiparticle concept, thermodynamic properties, instabilities within a Fermi-liquid description, finite-temperature corrections, transport properties, and the Kondo effect. It discusses the breakdown of the Fermi-liquid concept at quantum phase transitions, including classical vs. quantum phase transitions, scaling properties, itinerant fermion systems, thermodynamic quantities, self-consistent spin-fluctuation theories, transport properties, and the approach to the quantum critical point (QCP). The breakdown of the Hertz model of a magnetic QCP is analyzed, including multiple dynamical exponents, infinitely many marginal operators, self-energy effects, pseudogaps, itinerant antiferromagnetism, and superconductivity. The review also discusses the breakdown of the Kondo effect in heavy-fermion metals, including "local" QCP within extended DMFT, fractionalized Fermi liquid and deconfined criticality, spin-charge separation at the QCP, and one vs. two transitions. It addresses disorder effects close to quantum phase transitions, including Harris criterion and fixed points, rare regions and quantum Griffiths singularities, effects of rare regions on metallic QPT, and metallic quantum glasses. The review presents experimental systems where quantum criticality and non-Fermi-liquid (NFL) behavior have been observed, including heavy-fermion systems, itinerant transition-metal magnets, and superconductivity near the magnetic-non-magnetic quantum phase transition. The review concludes with a discussion of the implications of these findings for the understanding of Fermi-liquid instabilities at quantum phase transitions.This review discusses the breakdown of the Fermi-liquid state of conduction electrons in metals near magnetic quantum phase transitions. It presents existing theoretical concepts and experimental data on selected materials to assess the validity of current theory. After recalling Fermi-liquid theory, the review describes the local Fermi-liquid state in quantum impurity models and their lattice versions. Scaling concepts applicable to quantum phase transitions are then presented, with a detailed description of the Hertz-Millis-Moriya theory. The breakdown of this theory is analyzed in several examples. Experimental data on heavy-fermion materials and transition-metal alloys are reviewed and compared with existing theory. The review covers the fundamentals of Fermi-liquid theory, including the quasiparticle concept, thermodynamic properties, instabilities within a Fermi-liquid description, finite-temperature corrections, transport properties, and the Kondo effect. It discusses the breakdown of the Fermi-liquid concept at quantum phase transitions, including classical vs. quantum phase transitions, scaling properties, itinerant fermion systems, thermodynamic quantities, self-consistent spin-fluctuation theories, transport properties, and the approach to the quantum critical point (QCP). The breakdown of the Hertz model of a magnetic QCP is analyzed, including multiple dynamical exponents, infinitely many marginal operators, self-energy effects, pseudogaps, itinerant antiferromagnetism, and superconductivity. The review also discusses the breakdown of the Kondo effect in heavy-fermion metals, including "local" QCP within extended DMFT, fractionalized Fermi liquid and deconfined criticality, spin-charge separation at the QCP, and one vs. two transitions. It addresses disorder effects close to quantum phase transitions, including Harris criterion and fixed points, rare regions and quantum Griffiths singularities, effects of rare regions on metallic QPT, and metallic quantum glasses. The review presents experimental systems where quantum criticality and non-Fermi-liquid (NFL) behavior have been observed, including heavy-fermion systems, itinerant transition-metal magnets, and superconductivity near the magnetic-non-magnetic quantum phase transition. The review concludes with a discussion of the implications of these findings for the understanding of Fermi-liquid instabilities at quantum phase transitions.