Semiclassical theories of the anomalous Hall effect

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.