This paper investigates the edge states and topological invariants of non-Hermitian systems, focusing on the non-Hermitian skin effect and its implications for the bulk-boundary correspondence. The authors show that the non-Hermitian skin effect, where eigenstates are localized near the boundary, necessitates a redefinition of topological invariants in a generalized Brillouin zone. This leads to a non-Bloch bulk-boundary correspondence, where topological edge modes are determined by non-Bloch topological invariants rather than the conventional Bloch-Hamiltonian-based topological number. The study uses the non-Hermitian Su-Schrieffer-Heeger (SSH) model as a case study. The model exhibits a non-Hermitian skin effect, where eigenstates are localized at the ends of the chain. This effect is analyzed using both analytical and numerical methods, revealing that the topological edge modes are determined by a non-Bloch winding number. The non-Bloch winding number is defined on a generalized Brillouin zone and correctly predicts the number of topological edge modes. The paper also discusses the implications of the non-Hermitian skin effect for the bulk-boundary correspondence. It shows that the conventional bulk-boundary correspondence breaks down in non-Hermitian systems, and a new non-Bloch bulk-boundary correspondence emerges. This new correspondence is characterized by the non-Bloch topological invariant, which accounts for the non-Hermitian skin effect and determines the topological edge modes. The study highlights the importance of the non-Hermitian skin effect in non-Hermitian systems and its role in redefining the topological invariants. The results demonstrate that the non-Bloch bulk-boundary correspondence is a more accurate description of the topological properties of non-Hermitian systems, particularly in the presence of non-Hermitian Hamiltonians. The findings have implications for the understanding of topological phases in non-Hermitian systems and may guide future research in this area.This paper investigates the edge states and topological invariants of non-Hermitian systems, focusing on the non-Hermitian skin effect and its implications for the bulk-boundary correspondence. The authors show that the non-Hermitian skin effect, where eigenstates are localized near the boundary, necessitates a redefinition of topological invariants in a generalized Brillouin zone. This leads to a non-Bloch bulk-boundary correspondence, where topological edge modes are determined by non-Bloch topological invariants rather than the conventional Bloch-Hamiltonian-based topological number. The study uses the non-Hermitian Su-Schrieffer-Heeger (SSH) model as a case study. The model exhibits a non-Hermitian skin effect, where eigenstates are localized at the ends of the chain. This effect is analyzed using both analytical and numerical methods, revealing that the topological edge modes are determined by a non-Bloch winding number. The non-Bloch winding number is defined on a generalized Brillouin zone and correctly predicts the number of topological edge modes. The paper also discusses the implications of the non-Hermitian skin effect for the bulk-boundary correspondence. It shows that the conventional bulk-boundary correspondence breaks down in non-Hermitian systems, and a new non-Bloch bulk-boundary correspondence emerges. This new correspondence is characterized by the non-Bloch topological invariant, which accounts for the non-Hermitian skin effect and determines the topological edge modes. The study highlights the importance of the non-Hermitian skin effect in non-Hermitian systems and its role in redefining the topological invariants. The results demonstrate that the non-Bloch bulk-boundary correspondence is a more accurate description of the topological properties of non-Hermitian systems, particularly in the presence of non-Hermitian Hamiltonians. The findings have implications for the understanding of topological phases in non-Hermitian systems and may guide future research in this area.