The Berry phase, discovered by Michael Berry in 1984, has become a fundamental concept in physics, influencing various branches of the field. This review, authored by Di Xiao, Ming-Che Chang, and Qian Niu, provides a comprehensive overview of the Berry phase's impact on material properties and its applications in solid-state physics. The authors start with a brief introduction to the necessary background, including the Berry phase's gauge invariance, geometrical nature, and topological significance. They then delve into the Berry phase effect in various solid-state applications, focusing on the semiclassical formulation of electron dynamics, which is a versatile tool for studying electron behavior in electromagnetic fields and other perturbations. Key topics covered include: 1. **Adiabatic Transport and Electric Polarization**: The Berry phase is shown to be responsible for adiabatic current and quantized adiabatic particle transport in band insulators. The Berry curvature, a gauge-invariant quantity, plays a crucial role in these phenomena. 2. **Electron Dynamics in Electric Fields**: The Berry phase affects the transverse velocity of electrons, leading to Hall currents. The review discusses the quantum, anomalous, and valley Hall effects. 3. **Wave Packet Dynamics**: The construction and properties of electron wave packets are explored, including orbital magnetization and dipole moment. The Berry phase is shown to influence these properties. 4. **Electron Dynamics in Electromagnetic Fields**: The equations of motion and modified density of states are derived, demonstrating how the Berry phase modifies these quantities. 5. **General Perturbations**: The review covers electron dynamics under spatially varying perturbations, such as deformed crystals and inhomogeneous order parameters. 6. **Quantization of Electron Dynamics**: The Bohr-Sommerfeld quantization rule and the Wannier-Stark ladder are discussed, along with the quantization of electron dynamics in the presence of magnetic fields. 7. **Non-Abelian Formulation**: The Berry phase's role in non-Abelian Berry curvature and its applications in spin dynamics and relativistic systems is explored. The authors emphasize that the Berry phase is essential for understanding basic material properties and that it should be included as a fundamental ingredient in the theoretical framework. The review concludes with a discussion on the re-quantization of the semiclassical theory, providing a broader and more rigorous understanding of the Berry phase effects in solid-state physics.The Berry phase, discovered by Michael Berry in 1984, has become a fundamental concept in physics, influencing various branches of the field. This review, authored by Di Xiao, Ming-Che Chang, and Qian Niu, provides a comprehensive overview of the Berry phase's impact on material properties and its applications in solid-state physics. The authors start with a brief introduction to the necessary background, including the Berry phase's gauge invariance, geometrical nature, and topological significance. They then delve into the Berry phase effect in various solid-state applications, focusing on the semiclassical formulation of electron dynamics, which is a versatile tool for studying electron behavior in electromagnetic fields and other perturbations. Key topics covered include: 1. **Adiabatic Transport and Electric Polarization**: The Berry phase is shown to be responsible for adiabatic current and quantized adiabatic particle transport in band insulators. The Berry curvature, a gauge-invariant quantity, plays a crucial role in these phenomena. 2. **Electron Dynamics in Electric Fields**: The Berry phase affects the transverse velocity of electrons, leading to Hall currents. The review discusses the quantum, anomalous, and valley Hall effects. 3. **Wave Packet Dynamics**: The construction and properties of electron wave packets are explored, including orbital magnetization and dipole moment. The Berry phase is shown to influence these properties. 4. **Electron Dynamics in Electromagnetic Fields**: The equations of motion and modified density of states are derived, demonstrating how the Berry phase modifies these quantities. 5. **General Perturbations**: The review covers electron dynamics under spatially varying perturbations, such as deformed crystals and inhomogeneous order parameters. 6. **Quantization of Electron Dynamics**: The Bohr-Sommerfeld quantization rule and the Wannier-Stark ladder are discussed, along with the quantization of electron dynamics in the presence of magnetic fields. 7. **Non-Abelian Formulation**: The Berry phase's role in non-Abelian Berry curvature and its applications in spin dynamics and relativistic systems is explored. The authors emphasize that the Berry phase is essential for understanding basic material properties and that it should be included as a fundamental ingredient in the theoretical framework. The review concludes with a discussion on the re-quantization of the semiclassical theory, providing a broader and more rigorous understanding of the Berry phase effects in solid-state physics.