This paper presents a systematic framework for understanding topological phases in non-Hermitian systems. The authors explore the topological properties of non-Hermitian systems, which differ from their Hermitian counterparts in that they can exhibit unique phases with no Hermitian counterparts. The paper introduces a classification of non-Hermitian topological phases based on the Altland-Zirnbauer (AZ) classes, which are used to classify topological insulators and superconductors in Hermitian systems. The authors show that non-Hermitian systems can be classified using K-theory, leading to a periodic table of non-Hermitian topological phases. The paper also discusses the robustness of non-Hermitian topological phases against disorder and the existence of a bulk-edge correspondence in non-Hermitian systems. The authors demonstrate that non-Hermitian systems can support topological phases even in the absence of symmetry, and they identify a Z₂ topological index for arbitrary quantum channels. The paper concludes with a discussion of the implications of these findings for the broader understanding of topology in non-Hermitian systems.This paper presents a systematic framework for understanding topological phases in non-Hermitian systems. The authors explore the topological properties of non-Hermitian systems, which differ from their Hermitian counterparts in that they can exhibit unique phases with no Hermitian counterparts. The paper introduces a classification of non-Hermitian topological phases based on the Altland-Zirnbauer (AZ) classes, which are used to classify topological insulators and superconductors in Hermitian systems. The authors show that non-Hermitian systems can be classified using K-theory, leading to a periodic table of non-Hermitian topological phases. The paper also discusses the robustness of non-Hermitian topological phases against disorder and the existence of a bulk-edge correspondence in non-Hermitian systems. The authors demonstrate that non-Hermitian systems can support topological phases even in the absence of symmetry, and they identify a Z₂ topological index for arbitrary quantum channels. The paper concludes with a discussion of the implications of these findings for the broader understanding of topology in non-Hermitian systems.