This paper explores the topological phases of non-Hermitian systems, which are characterized by complex eigenvalues and lack the symmetry constraints of Hermitian systems. The authors develop a coherent framework to understand these phases, starting with one-dimensional lattices that exhibit topological phases with no Hermitian counterparts. They define a topological winding number, which is robust against disorder and can be measured from wave-packet dynamics. The paper also introduces a bulk-edge correspondence, where an infinite number of (quasi-)edge modes exist in semi-infinite systems. The authors apply K-theory to classify non-Hermitian topological phases in all dimensions, leading to a periodic table that unifies time-reversal and particle-hole symmetries. They identify a $\mathbb{Z}_2$ topological index for arbitrary quantum channels and provide concrete examples of non-Hermitian topological phases in zero and one dimensions. The work lays the foundation for a unified understanding of topology in non-Hermitian systems.This paper explores the topological phases of non-Hermitian systems, which are characterized by complex eigenvalues and lack the symmetry constraints of Hermitian systems. The authors develop a coherent framework to understand these phases, starting with one-dimensional lattices that exhibit topological phases with no Hermitian counterparts. They define a topological winding number, which is robust against disorder and can be measured from wave-packet dynamics. The paper also introduces a bulk-edge correspondence, where an infinite number of (quasi-)edge modes exist in semi-infinite systems. The authors apply K-theory to classify non-Hermitian topological phases in all dimensions, leading to a periodic table that unifies time-reversal and particle-hole symmetries. They identify a $\mathbb{Z}_2$ topological index for arbitrary quantum channels and provide concrete examples of non-Hermitian topological phases in zero and one dimensions. The work lays the foundation for a unified understanding of topology in non-Hermitian systems.