The article provides a comprehensive review of topological band theory, emphasizing its role in discovering new topologically interesting materials and understanding the characteristics of topological states. It begins by discussing the foundational aspects of topological band theory, including methods for evaluating topological invariants, such as the Z₂ invariant, and the computation of surface/edge states. The authors highlight the importance of spin-orbit coupling (SOC) in creating topological states and the experimental verification of these states in various materials. The review then delves into other topological states of quantum matter, such as topological crystalline insulators (TCIs), disorder or interaction-driven topological insulators (TIs), topological superconductors, Weyl and 3D Dirac semimetals, and topological phase transitions. It discusses the experimental verification of theoretically predicted properties and protections of topological states, including the observation of Dirac cones, helical spin textures, and suppression of backscattering. A survey of currently predicted topological materials is provided, covering binary, ternary, and quaternary compounds, transition metal and f-electron materials, complex oxides, organometallics, skutterudites, and antiperovskites. The article also explores the emerging area of 2D atomically thin films and their potential applications. Finally, the authors outline successful strategies for new materials discovery and conclude with perspectives on future research directions in the field of topological materials.The article provides a comprehensive review of topological band theory, emphasizing its role in discovering new topologically interesting materials and understanding the characteristics of topological states. It begins by discussing the foundational aspects of topological band theory, including methods for evaluating topological invariants, such as the Z₂ invariant, and the computation of surface/edge states. The authors highlight the importance of spin-orbit coupling (SOC) in creating topological states and the experimental verification of these states in various materials. The review then delves into other topological states of quantum matter, such as topological crystalline insulators (TCIs), disorder or interaction-driven topological insulators (TIs), topological superconductors, Weyl and 3D Dirac semimetals, and topological phase transitions. It discusses the experimental verification of theoretically predicted properties and protections of topological states, including the observation of Dirac cones, helical spin textures, and suppression of backscattering. A survey of currently predicted topological materials is provided, covering binary, ternary, and quaternary compounds, transition metal and f-electron materials, complex oxides, organometallics, skutterudites, and antiperovskites. The article also explores the emerging area of 2D atomically thin films and their potential applications. Finally, the authors outline successful strategies for new materials discovery and conclude with perspectives on future research directions in the field of topological materials.