  Quantum criticality describes the collective fluctuations of matter
undergoing a second-order phase transition at zero temperature. It is being
discussed in a number of strongly correlated electron systems. A prototype case
occurs in the heavy fermion metals, in which antiferromagnetic quantum critical
points have been explicitly observed. Here, I address two types of
antiferromagnetic quantum critical points. In addition to the standard
description based on the fluctuations of the antiferromagnetic order, a local
quantum critical point is also considered. It contains inherently quantum modes
that are associated with a critical breakdown of the Kondo effect. Across such
a quantum critical point, there is a sudden collapse of a large Fermi surface
to a small one. I also consider the proximate antiferromagnetic and
paramagnetic phases, and these considerations lead to a global phase diagram.
Finally, I discuss the pertinent experiments on the antiferromagnetic heavy
fermions, briefly address the case of ferromagnetic heavy fermions, and outline
some directions for future studies.
