The review by Nagaosa, Sinova, Onoda, MacDonald, and Ong provides a comprehensive overview of the anomalous Hall effect (AHE), focusing on recent developments that have clarified the understanding of this phenomenon. The AHE, which involves subtle quantum mechanical and topological effects, has been a subject of intense study for nearly a century due to its complex nature. The review highlights the synergy between experimental and theoretical efforts, which have been crucial in advancing the field. **Theoretical Advances:** - **Berry Phase Concepts:** The adoption of Berry phase concepts has linked the AHE to the topological nature of Hall currents originating from spin-orbit coupling. This connection has provided a deeper understanding of the intrinsic AHE in metallic ferromagnets with moderate conductivity. - **Topological Interpretation:** The intrinsic AHE is now understood as a quantum mechanical property of a perfect crystal, expressed in terms of Berry phase curvatures. This mechanism is dominant in metallic ferromagnets with moderate conductivity. **Experimental Developments:** - **New Studies in Various Materials:** Experimental studies in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors have established systematic trends. These studies have helped in understanding the AHE in different material classes. - **Systematic Trends:** The AHE in these materials exhibits distinct regimes characterized by the conductivity $\sigma_{xx}$, with the skew scattering contribution dominating in high conductivity regimes, the intrinsic and side-jump contributions dominating in good metal regimes, and the bad metal/hopping regime showing a dependence on $\sigma_{xx}^{1.6-1.8}$. **Theoretical Treatments:** - **Semiclassical and Quantum-Mechanical Approaches:** The review discusses both semiclassical and quantum-mechanical treatments of the AHE, including the Kubo and Keldysh formalisms. It emphasizes the equivalence of these theories in the metallic regime and the importance of considering multiband effects. **Future Directions:** - **Global View:** The review proposes a tentative global view of the AHE, summarizing the roles of intrinsic and extrinsic contributions in the disorder-strength vs. temperature plane. - **Outstanding Issues:** The authors highlight several intriguing questions and avenues for future research, emphasizing the need for further experimental and theoretical investigations. Overall, the review aims to provide a comprehensive and systematic understanding of the AHE, bridging the gap between experimental observations and theoretical explanations.The review by Nagaosa, Sinova, Onoda, MacDonald, and Ong provides a comprehensive overview of the anomalous Hall effect (AHE), focusing on recent developments that have clarified the understanding of this phenomenon. The AHE, which involves subtle quantum mechanical and topological effects, has been a subject of intense study for nearly a century due to its complex nature. The review highlights the synergy between experimental and theoretical efforts, which have been crucial in advancing the field. **Theoretical Advances:** - **Berry Phase Concepts:** The adoption of Berry phase concepts has linked the AHE to the topological nature of Hall currents originating from spin-orbit coupling. This connection has provided a deeper understanding of the intrinsic AHE in metallic ferromagnets with moderate conductivity. - **Topological Interpretation:** The intrinsic AHE is now understood as a quantum mechanical property of a perfect crystal, expressed in terms of Berry phase curvatures. This mechanism is dominant in metallic ferromagnets with moderate conductivity. **Experimental Developments:** - **New Studies in Various Materials:** Experimental studies in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors have established systematic trends. These studies have helped in understanding the AHE in different material classes. - **Systematic Trends:** The AHE in these materials exhibits distinct regimes characterized by the conductivity $\sigma_{xx}$, with the skew scattering contribution dominating in high conductivity regimes, the intrinsic and side-jump contributions dominating in good metal regimes, and the bad metal/hopping regime showing a dependence on $\sigma_{xx}^{1.6-1.8}$. **Theoretical Treatments:** - **Semiclassical and Quantum-Mechanical Approaches:** The review discusses both semiclassical and quantum-mechanical treatments of the AHE, including the Kubo and Keldysh formalisms. It emphasizes the equivalence of these theories in the metallic regime and the importance of considering multiband effects. **Future Directions:** - **Global View:** The review proposes a tentative global view of the AHE, summarizing the roles of intrinsic and extrinsic contributions in the disorder-strength vs. temperature plane. - **Outstanding Issues:** The authors highlight several intriguing questions and avenues for future research, emphasizing the need for further experimental and theoretical investigations. Overall, the review aims to provide a comprehensive and systematic understanding of the AHE, bridging the gap between experimental observations and theoretical explanations.