This review discusses the development and application of topological band theory in understanding and predicting topological materials. The theory provides a framework for analyzing the topological properties of electronic band structures, which are crucial for identifying and characterizing topological states of matter. The review begins by outlining the foundational concepts of topological band theory, including the computation of topological invariants such as the Z2 invariant, which characterizes the topological nature of materials. It discusses methods for evaluating these invariants, including systems without inversion symmetry and interacting systems, and highlights the role of symmetry in protecting topological states. The review then explores various types of topological states, including topological crystalline insulators, disorder or interaction-driven topological insulators, topological superconductors, Weyl and 3D Dirac semimetals, and topological phase transitions. It provides a comprehensive survey of currently predicted 2D and 3D topological materials, covering a wide range of compounds and structures, including binary, ternary, and quaternary materials, transition metal and f-electron systems, and complex oxides. The review also discusses the potential applications of these materials in various technologies, such as thermoelectrics, spintronics, and quantum computing. The review emphasizes the importance of first-principles band theory in predicting new topological materials and the experimental verification of their properties. It highlights the role of computational methods in understanding the electronic structure and topological properties of materials, as well as the challenges in modeling these systems within a first-principles framework. The review concludes with a perspective on future research directions in the field of topological materials, emphasizing the need for further exploration of the interplay between strong electron correlations and spin-orbit coupling in the discovery of new topological phases.This review discusses the development and application of topological band theory in understanding and predicting topological materials. The theory provides a framework for analyzing the topological properties of electronic band structures, which are crucial for identifying and characterizing topological states of matter. The review begins by outlining the foundational concepts of topological band theory, including the computation of topological invariants such as the Z2 invariant, which characterizes the topological nature of materials. It discusses methods for evaluating these invariants, including systems without inversion symmetry and interacting systems, and highlights the role of symmetry in protecting topological states. The review then explores various types of topological states, including topological crystalline insulators, disorder or interaction-driven topological insulators, topological superconductors, Weyl and 3D Dirac semimetals, and topological phase transitions. It provides a comprehensive survey of currently predicted 2D and 3D topological materials, covering a wide range of compounds and structures, including binary, ternary, and quaternary materials, transition metal and f-electron systems, and complex oxides. The review also discusses the potential applications of these materials in various technologies, such as thermoelectrics, spintronics, and quantum computing. The review emphasizes the importance of first-principles band theory in predicting new topological materials and the experimental verification of their properties. It highlights the role of computational methods in understanding the electronic structure and topological properties of materials, as well as the challenges in modeling these systems within a first-principles framework. The review concludes with a perspective on future research directions in the field of topological materials, emphasizing the need for further exploration of the interplay between strong electron correlations and spin-orbit coupling in the discovery of new topological phases.