  The Hall effect, in which current flows perpendicular to an applied
electrical bias, has played a prominent role in modern condensed matter physics
over much of the subject's history. Appearing variously in classical,
relativistic and quantum guises, it has among other roles contributed to
establishing the band theory of solids, to research on new phases of strongly
interacting electrons, and to the phenomenology of topological condensed
matter. The dissipationless Hall current requires time-reversal symmetry
breaking. For over a century it has either been ascribed to externally applied
magnetic field and referred to as the ordinary Hall effect, or ascribed to
spontaneous non-zero total internal magnetization (ferromagnetism) and referred
to as the anomalous Hall effect. It has not commonly been associated with
antiferromagnetic order. More recently, however, theoretical predictions and
experimental observations have identified large Hall effects in some
compensated magnetic crystals, governed by neither of the global
magnetic-dipole symmetry breaking mechanisms mentioned above. The goals of this
article are to systematically organize the present understanding of anomalous
antiferromagnetic materials that generate a non-zero Hall effect, which we will
call anomalous Hall antiferromagnets, and to discuss this class of materials in
a broader fundamental and applied research context. Our motivation in drawing
attention to anomalous Hall antiferromagnets is two-fold. First, since Hall
effects that are not governed by magnetic dipole symmetry breaking are at odds
with the traditional understanding of the phenomenon, the topic deserves
attention on its own. Second, this new reincarnation has again placed the Hall
effect in the middle of an emerging field of physics at the intersection of
multipole magnetism, topological condensed matter, and spintronics.
