Topological insulators and superconductors are gapped phases of non-interacting fermions with topologically protected boundary modes. These phases are classified into five distinct classes based on symmetry and spatial dimension. The classification is determined by topological invariants, such as integer Chern numbers or Z₂ quantities. The ten-fold way, a classification of Hamiltonians based on symmetry, includes ten symmetry classes, derived from the work of Zirnbauer, Altland, and Zirnbauer. These classes are related to symmetric spaces and their properties, reflecting the 8-fold periodicity of spinor representations of orthogonal groups. The paper discusses how topological insulators and superconductors in different dimensions and classes can be related via dimensional reduction, where compactifying spatial dimensions reduces the system to a lower dimension. Z-topological insulators descend by one dimension, while Z₂-topological insulators are lower-dimensional descendants of Z-topological insulators. The paper also explores the relationship between topological invariants and Chern-Simons invariants, and discusses topological field theories for linear responses in topological insulators. It also addresses how inversion symmetry affects the classification of topological insulators and superconductors. The classification is summarized in table 3, showing five classes in each spatial dimension, with three characterized by integer invariants and two by Z₂ invariants. The paper provides examples of terms of topological origin, such as WZW and Z₂ terms, and discusses their role in Anderson localization. The classification is further supported by the periodicity of homotopy groups and the connection to K-theory. The paper concludes with a discussion of weak topological insulators and superconductors, and the existence of zero modes on topological defects.Topological insulators and superconductors are gapped phases of non-interacting fermions with topologically protected boundary modes. These phases are classified into five distinct classes based on symmetry and spatial dimension. The classification is determined by topological invariants, such as integer Chern numbers or Z₂ quantities. The ten-fold way, a classification of Hamiltonians based on symmetry, includes ten symmetry classes, derived from the work of Zirnbauer, Altland, and Zirnbauer. These classes are related to symmetric spaces and their properties, reflecting the 8-fold periodicity of spinor representations of orthogonal groups. The paper discusses how topological insulators and superconductors in different dimensions and classes can be related via dimensional reduction, where compactifying spatial dimensions reduces the system to a lower dimension. Z-topological insulators descend by one dimension, while Z₂-topological insulators are lower-dimensional descendants of Z-topological insulators. The paper also explores the relationship between topological invariants and Chern-Simons invariants, and discusses topological field theories for linear responses in topological insulators. It also addresses how inversion symmetry affects the classification of topological insulators and superconductors. The classification is summarized in table 3, showing five classes in each spatial dimension, with three characterized by integer invariants and two by Z₂ invariants. The paper provides examples of terms of topological origin, such as WZW and Z₂ terms, and discusses their role in Anderson localization. The classification is further supported by the periodicity of homotopy groups and the connection to K-theory. The paper concludes with a discussion of weak topological insulators and superconductors, and the existence of zero modes on topological defects.