  We present a review of experimental and theoretical studies of the anomalous
Hall effect (AHE), focusing on recent developments that have provided a more
complete framework for understanding this subtle phenomenon and have, in many
instances, replaced controversy by clarity. Synergy between experimental and
theoretical work, both playing a crucial role, has been at the heart of these
advances. On the theoretical front, the adoption of Berry-phase concepts has
established a link between the AHE and the topological nature of the Hall
currents which originate from spin-orbit coupling. On the experimental front,
new experimental studies of the AHE in transition metals, transition-metal
oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have
more clearly established systematic trends. These two developments in concert
with first-principles electronic structure calculations, strongly favor the
dominance of an intrinsic Berry-phase-related AHE mechanism in metallic
ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in
terms of Berry-phase curvatures and it is therefore an intrinsic quantum
mechanical property of a perfect cyrstal. An extrinsic mechanism, skew
scattering from disorder, tends to dominate the AHE in highly conductive
ferromagnets. We review the full modern semiclassical treatment of the AHE
together with the more rigorous quantum-mechanical treatments based on the Kubo
and Keldysh formalisms, taking into account multiband effects, and demonstrate
the equivalence of all three linear response theories in the metallic regime.
Finally we discuss outstanding issues and avenues for future investigation.
