  This review discusses instabilities of the Fermi-liquid state of conduction
electrons in metals with particular emphasis on magnetic quantum critical
points. Both the existing theoretical concepts and experimental data on
selected materials are presented; with the aim of assessing the validity of
presently available theory. After briefly recalling the fundamentals of
Fermi-liquid theory, the local Fermi-liquid state in quantum impurity models
and their lattice versions is described. Next, the scaling concepts applicable
to quantum phase transitions are presented. The Hertz-Millis-Moriya theory of
quantum phase transitions is described in detail. The breakdown of the latter
is analyzed in several examples. In the final part experimental data on
heavy-fermion materials and transition-metal alloys are reviewed and confronted
with existing theory.
