Topological insulators and superconductors are new quantum states of matter that cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating bulk gap and protected gapless edge or surface states due to time-reversal (TR) symmetry. These materials have been theoretically predicted and experimentally observed in systems such as HgTe quantum wells, BiSb alloys, and Bi₂Te₃ and Bi₂Se₃ crystals. The review discusses theoretical models, materials properties, and experimental results for two-dimensional (2D) and three-dimensional (3D) topological insulators, as well as topological superconductors, which have a full pairing gap and gapless surface states consisting of Majorana fermions. The topological classification of these materials is based on the concept of topological invariants, which are robust against small perturbations. The 2D topological insulator, also known as the quantum spin Hall (QSH) insulator, was first predicted in 2006 and experimentally observed in HgTe/CdTe quantum wells. It has protected helical edge states with opposite spin polarizations counter-propagating along the edge. The 3D topological insulator was predicted in Bi₁₋ₓSbₓ alloys and observed in Bi₂Te₃, Sb₂Te₃, and Bi₂Se₃ crystals. These materials have protected surface states with a single Dirac cone. The topological properties of these materials are described by the Z₂ topological invariant, which classifies them into two distinct classes: trivial and non-trivial. The non-trivial class has a full insulating bulk gap but gapless edge or surface states. The topological protection of these states is due to TR symmetry, which prevents backscattering and ensures robustness against impurities and disorder. The review also discusses the general theory of topological insulators, including topological field theory (TFT) and topological band theory (TBT). These theories provide a framework for understanding the topological properties of these materials and predicting experimentally measurable effects such as the topological magnetoelectric effect. The topological superconductors, which are closely related to topological insulators, have Majorana zero modes at their surfaces, which are important for quantum computing applications. The review highlights the importance of topological insulators and superconductors in condensed matter physics, as they represent new states of matter with unique properties that are robust against perturbations. These materials have potential applications in quantum computing, spintronics, and other areas of technology. The study of topological insulators and superconductors continues to be an active area of research, with new discoveries and theoretical developments expanding our understanding of these fascinating quantum states of matter.Topological insulators and superconductors are new quantum states of matter that cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating bulk gap and protected gapless edge or surface states due to time-reversal (TR) symmetry. These materials have been theoretically predicted and experimentally observed in systems such as HgTe quantum wells, BiSb alloys, and Bi₂Te₃ and Bi₂Se₃ crystals. The review discusses theoretical models, materials properties, and experimental results for two-dimensional (2D) and three-dimensional (3D) topological insulators, as well as topological superconductors, which have a full pairing gap and gapless surface states consisting of Majorana fermions. The topological classification of these materials is based on the concept of topological invariants, which are robust against small perturbations. The 2D topological insulator, also known as the quantum spin Hall (QSH) insulator, was first predicted in 2006 and experimentally observed in HgTe/CdTe quantum wells. It has protected helical edge states with opposite spin polarizations counter-propagating along the edge. The 3D topological insulator was predicted in Bi₁₋ₓSbₓ alloys and observed in Bi₂Te₃, Sb₂Te₃, and Bi₂Se₃ crystals. These materials have protected surface states with a single Dirac cone. The topological properties of these materials are described by the Z₂ topological invariant, which classifies them into two distinct classes: trivial and non-trivial. The non-trivial class has a full insulating bulk gap but gapless edge or surface states. The topological protection of these states is due to TR symmetry, which prevents backscattering and ensures robustness against impurities and disorder. The review also discusses the general theory of topological insulators, including topological field theory (TFT) and topological band theory (TBT). These theories provide a framework for understanding the topological properties of these materials and predicting experimentally measurable effects such as the topological magnetoelectric effect. The topological superconductors, which are closely related to topological insulators, have Majorana zero modes at their surfaces, which are important for quantum computing applications. The review highlights the importance of topological insulators and superconductors in condensed matter physics, as they represent new states of matter with unique properties that are robust against perturbations. These materials have potential applications in quantum computing, spintronics, and other areas of technology. The study of topological insulators and superconductors continues to be an active area of research, with new discoveries and theoretical developments expanding our understanding of these fascinating quantum states of matter.