This paper presents a comprehensive theory of symmetry and topology in non-Hermitian physics, extending the well-known Altland-Zirnbauer (AZ) symmetry classification for Hermitian systems. Non-Hermitian systems, which lack Hermiticity, exhibit unique properties such as non-orthogonal eigenstates and complex-energy spectra, leading to two types of complex-energy gaps: point-like and line-like. The authors demonstrate that non-Hermiticity modifies the nature of symmetries, particularly charge conjugation and chiral symmetry, and introduces a new symmetry class, pseudo-Hermiticity, which is essential for non-Hermitian systems. The total number of symmetry classes is found to be 38, significantly more than the 10 AZ classes in Hermitian systems. This classification includes both intrinsic non-Hermitian topological phases and non-Hermitian random matrices. The authors also show that non-Hermitian topology depends on the type of complex-energy gap, leading to multiple topological structures for each symmetry class and spatial dimension. The bulk-boundary correspondence in non-Hermitian systems is clarified, and symmetries that prevent the non-Hermitian skin effect are identified. The theory is applied to predict new types of symmetry-protected topological lasers and dissipative topological superconductors, and provides a topological classification for both Hermitian and non-Hermitian free bosons. The work establishes a theoretical framework for understanding non-Hermitian topological phases and paves the way for uncovering unique phenomena and functionalities arising from the interplay of non-Hermiticity and topology.This paper presents a comprehensive theory of symmetry and topology in non-Hermitian physics, extending the well-known Altland-Zirnbauer (AZ) symmetry classification for Hermitian systems. Non-Hermitian systems, which lack Hermiticity, exhibit unique properties such as non-orthogonal eigenstates and complex-energy spectra, leading to two types of complex-energy gaps: point-like and line-like. The authors demonstrate that non-Hermiticity modifies the nature of symmetries, particularly charge conjugation and chiral symmetry, and introduces a new symmetry class, pseudo-Hermiticity, which is essential for non-Hermitian systems. The total number of symmetry classes is found to be 38, significantly more than the 10 AZ classes in Hermitian systems. This classification includes both intrinsic non-Hermitian topological phases and non-Hermitian random matrices. The authors also show that non-Hermitian topology depends on the type of complex-energy gap, leading to multiple topological structures for each symmetry class and spatial dimension. The bulk-boundary correspondence in non-Hermitian systems is clarified, and symmetries that prevent the non-Hermitian skin effect are identified. The theory is applied to predict new types of symmetry-protected topological lasers and dissipative topological superconductors, and provides a topological classification for both Hermitian and non-Hermitian free bosons. The work establishes a theoretical framework for understanding non-Hermitian topological phases and paves the way for uncovering unique phenomena and functionalities arising from the interplay of non-Hermiticity and topology.