This paper classifies topological insulators and superconductors in three spatial dimensions (3D) based on symmetry classes derived from random matrix theory. The authors identify five out of ten symmetry classes that support topologically non-trivial insulators or superconductors. These include the well-known Z₂ topological insulator in the symplectic (or spin-orbit) symmetry class, as well as four additional topological phases. All these phases are time-reversal invariant and exhibit distinct topological sectors separated by quantum phase transitions. The topological properties are characterized by integer winding numbers in momentum space. In 3D topological insulators, the surface supports gapless Dirac or Majorana fermion modes that are robust against arbitrary perturbations, including disorder. These surface modes are protected by the bulk topological properties and are analogous to the quantum spin Hall effect in two dimensions. The paper also discusses the connection between the bulk topological properties and the surface modes, showing that the topological nature of the bulk is reflected in the surface physics. The authors use two complementary strategies to classify these topological phases: (1) introducing a topological invariant (winding number) to characterize the bulk ground state wave functions, and (2) studying the appearance of gapless surface modes at the boundary of 3D topological insulators. The surface Dirac fermion modes are shown to be robust against perturbations and are protected by the symmetries of the system. The paper also discusses the connection between these topological phases and well-known paired topological phases in two dimensions, such as the spinless chiral (p_x ± ip_y)-wave superconductor. The topological properties of the 3D phases are shown to be analogous to those of the 2D Moore-Read Pfaffian state. The paper concludes with a discussion of the topological degeneracies that arise in these 3D phases and their connection to Anderson localization physics at the surface.This paper classifies topological insulators and superconductors in three spatial dimensions (3D) based on symmetry classes derived from random matrix theory. The authors identify five out of ten symmetry classes that support topologically non-trivial insulators or superconductors. These include the well-known Z₂ topological insulator in the symplectic (or spin-orbit) symmetry class, as well as four additional topological phases. All these phases are time-reversal invariant and exhibit distinct topological sectors separated by quantum phase transitions. The topological properties are characterized by integer winding numbers in momentum space. In 3D topological insulators, the surface supports gapless Dirac or Majorana fermion modes that are robust against arbitrary perturbations, including disorder. These surface modes are protected by the bulk topological properties and are analogous to the quantum spin Hall effect in two dimensions. The paper also discusses the connection between the bulk topological properties and the surface modes, showing that the topological nature of the bulk is reflected in the surface physics. The authors use two complementary strategies to classify these topological phases: (1) introducing a topological invariant (winding number) to characterize the bulk ground state wave functions, and (2) studying the appearance of gapless surface modes at the boundary of 3D topological insulators. The surface Dirac fermion modes are shown to be robust against perturbations and are protected by the symmetries of the system. The paper also discusses the connection between these topological phases and well-known paired topological phases in two dimensions, such as the spinless chiral (p_x ± ip_y)-wave superconductor. The topological properties of the 3D phases are shown to be analogous to those of the 2D Moore-Read Pfaffian state. The paper concludes with a discussion of the topological degeneracies that arise in these 3D phases and their connection to Anderson localization physics at the surface.