Topological insulators and superconductors are novel states of quantum matter characterized by a full insulating gap in the bulk and gapless edge or surface states protected by time-reversal symmetry. These materials have been theoretically predicted and experimentally observed in various systems, including HgTe quantum wells, BiSb alloys, and Bi$_2$Te$_3$ and Bi$_2$Se$_3$ crystals. The review covers theoretical models, material properties, and experimental results for two-dimensional and three-dimensional topological insulators, discussing both topological band theory and topological field theory. Topological superconductors, which have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions, are also reviewed, drawing parallels with topological insulators. The article provides a comprehensive overview of the field, highlighting the key concepts and recent advancements in the study of these quantum states of matter.Topological insulators and superconductors are novel states of quantum matter characterized by a full insulating gap in the bulk and gapless edge or surface states protected by time-reversal symmetry. These materials have been theoretically predicted and experimentally observed in various systems, including HgTe quantum wells, BiSb alloys, and Bi$_2$Te$_3$ and Bi$_2$Se$_3$ crystals. The review covers theoretical models, material properties, and experimental results for two-dimensional and three-dimensional topological insulators, discussing both topological band theory and topological field theory. Topological superconductors, which have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions, are also reviewed, drawing parallels with topological insulators. The article provides a comprehensive overview of the field, highlighting the key concepts and recent advancements in the study of these quantum states of matter.