The paper discusses the classification of topological insulators and superconductors in various spatial dimensions, focusing on the ten-fold way and dimensional hierarchy. It introduces the concept of topological sectors within each class, distinguished by either a $\mathbb{Z}$ or $\mathbb{Z}_2$ topological invariant. The authors construct representatives of these topological phases using Dirac Hamiltonians and demonstrate how different dimensions and classes can be related through "dimensional reduction" by compactifying spatial dimensions. For $\mathbb{Z}$-topological phases, this involves descending by one dimension at a time, while $\mathbb{Z}_2$-topological phases are shown to be lower-dimensional descendants of parent $\mathbb{Z}$-topological phases. The 8-fold periodicity in dimension $d$ for topological phases with Hamiltonians satisfying reality conditions is linked to the 8-fold periodicity of spinor representations of orthogonal groups. The paper also derives a relation between the topological invariant characterizing topological phases with chiral symmetry (the winding number) and the Chern-Simons invariant. Additionally, it explores the impact of inversion symmetry on the classification of topological phases and discusses topological field theories describing linear responses in these systems.The paper discusses the classification of topological insulators and superconductors in various spatial dimensions, focusing on the ten-fold way and dimensional hierarchy. It introduces the concept of topological sectors within each class, distinguished by either a $\mathbb{Z}$ or $\mathbb{Z}_2$ topological invariant. The authors construct representatives of these topological phases using Dirac Hamiltonians and demonstrate how different dimensions and classes can be related through "dimensional reduction" by compactifying spatial dimensions. For $\mathbb{Z}$-topological phases, this involves descending by one dimension at a time, while $\mathbb{Z}_2$-topological phases are shown to be lower-dimensional descendants of parent $\mathbb{Z}$-topological phases. The 8-fold periodicity in dimension $d$ for topological phases with Hamiltonians satisfying reality conditions is linked to the 8-fold periodicity of spinor representations of orthogonal groups. The paper also derives a relation between the topological invariant characterizing topological phases with chiral symmetry (the winding number) and the Chern-Simons invariant. Additionally, it explores the impact of inversion symmetry on the classification of topological phases and discusses topological field theories describing linear responses in these systems.