The paper develops a comprehensive theory of symmetry and topology in non-Hermitian physics, addressing the unique challenges posed by non-Hermiticity. Non-Hermiticity enriches the topological phases beyond the Hermitian framework, leading to new symmetries and topological structures. The authors demonstrate that non-Hermiticity modifies the Altland-Zirnbauer (AZ) symmetry classification for insulators and superconductors, defining charge conjugation in terms of transposition rather than complex conjugation. This distinction also separates chiral symmetry from sublattice symmetry. The total number of symmetry classes is 38, including pseudo-Hermiticity, which provides a novel topological feature. The paper introduces two types of complex-energy gaps—point-like and line-like—each with distinct topological implications. Using these concepts and K-theory, the authors complete the classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. They show that non-Hermitian topology depends on the type of complex-energy gaps, with multiple topological structures appearing for each symmetry class and spatial dimension. The bulk-boundary correspondence in non-Hermitian systems is elucidated, and symmetries preventing the non-Hermitian skin effect are identified. The classification not only categorizes recent experimental observations but also predicts new types of symmetry-protected topological lasers and dissipative topological superconductors. The theory also provides a topological classification for both Hermitian and non-Hermitian free bosons, establishing a fundamental framework for understanding non-Hermitian topological phases.The paper develops a comprehensive theory of symmetry and topology in non-Hermitian physics, addressing the unique challenges posed by non-Hermiticity. Non-Hermiticity enriches the topological phases beyond the Hermitian framework, leading to new symmetries and topological structures. The authors demonstrate that non-Hermiticity modifies the Altland-Zirnbauer (AZ) symmetry classification for insulators and superconductors, defining charge conjugation in terms of transposition rather than complex conjugation. This distinction also separates chiral symmetry from sublattice symmetry. The total number of symmetry classes is 38, including pseudo-Hermiticity, which provides a novel topological feature. The paper introduces two types of complex-energy gaps—point-like and line-like—each with distinct topological implications. Using these concepts and K-theory, the authors complete the classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. They show that non-Hermitian topology depends on the type of complex-energy gaps, with multiple topological structures appearing for each symmetry class and spatial dimension. The bulk-boundary correspondence in non-Hermitian systems is elucidated, and symmetries preventing the non-Hermitian skin effect are identified. The classification not only categorizes recent experimental observations but also predicts new types of symmetry-protected topological lasers and dissipative topological superconductors. The theory also provides a topological classification for both Hermitian and non-Hermitian free bosons, establishing a fundamental framework for understanding non-Hermitian topological phases.