This paper presents a comprehensive framework for understanding the biorthogonal bulk-boundary correspondence in non-Hermitian systems. In non-Hermitian systems, the conventional bulk-boundary correspondence fails, as boundary states can appear or disappear at parameter values far from gap closings. The authors introduce the concept of biorthogonal quantum mechanics, where left and right eigenstates are individually decoupled from bulk physics, but their combined biorthogonal density penetrates the bulk at phase transitions. This leads to a generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is defined directly in systems with open boundaries. The authors illustrate their findings by deriving the phase diagram for several open boundary models, including non-Hermitian extensions of the Su-Schrieffer-Heeger (SSH) model and Chern insulators. They show that the biorthogonal polarization P accurately predicts the occurrence of boundary and domain-wall modes in a broad class of non-Hermitian systems, even in the presence of the non-Hermitian skin effect. The biorthogonal polarization P is defined as a quantized jump at the points where the number and/or localization properties of boundary modes change. The paper also discusses the stability of open boundary physics and the behavior of domain wall zero-modes. The authors show that the biorthogonal polarization P can be used to predict the occurrence of boundary modes in systems with open boundaries, and that the transitions in the open system are generally decoupled from the individual properties of the left and right eigenstates, leading to strikingly different phase diagrams. The authors conclude that the biorthogonal polarization provides a universal quantity that accurately predicts bulk transitions in systems with open boundaries associated with the (dis)appearance of boundary modes. This framework offers a new perspective on the bulk-boundary correspondence in non-Hermitian systems and provides a deeper understanding of the topological properties of such systems.This paper presents a comprehensive framework for understanding the biorthogonal bulk-boundary correspondence in non-Hermitian systems. In non-Hermitian systems, the conventional bulk-boundary correspondence fails, as boundary states can appear or disappear at parameter values far from gap closings. The authors introduce the concept of biorthogonal quantum mechanics, where left and right eigenstates are individually decoupled from bulk physics, but their combined biorthogonal density penetrates the bulk at phase transitions. This leads to a generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is defined directly in systems with open boundaries. The authors illustrate their findings by deriving the phase diagram for several open boundary models, including non-Hermitian extensions of the Su-Schrieffer-Heeger (SSH) model and Chern insulators. They show that the biorthogonal polarization P accurately predicts the occurrence of boundary and domain-wall modes in a broad class of non-Hermitian systems, even in the presence of the non-Hermitian skin effect. The biorthogonal polarization P is defined as a quantized jump at the points where the number and/or localization properties of boundary modes change. The paper also discusses the stability of open boundary physics and the behavior of domain wall zero-modes. The authors show that the biorthogonal polarization P can be used to predict the occurrence of boundary modes in systems with open boundaries, and that the transitions in the open system are generally decoupled from the individual properties of the left and right eigenstates, leading to strikingly different phase diagrams. The authors conclude that the biorthogonal polarization provides a universal quantity that accurately predicts bulk transitions in systems with open boundaries associated with the (dis)appearance of boundary modes. This framework offers a new perspective on the bulk-boundary correspondence in non-Hermitian systems and provides a deeper understanding of the topological properties of such systems.