  Recently, the semiclassical theory of the anomalous Hall effect induced by
the Berry curvature in Bloch bands has been introduced. The theory operates
only with gauge invariant concepts, that have a simple semiclassical
interpretation and provides a clear distinction among various contributions to
the Hall current. While the construction of such an approach to the anomalous
Hall effect problem has been long sought, only the new semiclassical theory
demonstrated the agreement with quantitative results of rigorous approaches
based on the Green function techniques. The purpose of this work is to review
the semiclassical approach including the early ideas and the recent
achievements.
