In topological insulators, the bulk-boundary correspondence describes the relationship between the bulk invariant -- computed for a system with periodic boundary conditions -- and the number of topological boundary states in the corresponding system with open boundary conditions. This is a well-known property of these systems and is important for predicting how they will behave. In recent years, however, the modeling of dissipative and non-equilibrium systems using non-Hermitian Hamiltonians has become increasingly popular. These systems feature many novel phenomena; in particular the bulk-boundary correspondence breaks down since the spectrum of the system with periodic boundary conditions typically differs fundamentally from the spectrum of the system with open boundary conditions.        

In this thesis, the behavior of the boundary states in non-Hermitian lattice models is studied. The framework of biorthogonal quantum mechanics is used to develop the biorthogonal bulk-boundary correspondence, which predicts the (dis)appearance of the boundary states in these systems. Closely related to the drastic change in spectra between boundary conditions is the non-Hermitian skin effect. This refers to the exponential localization of almost all eigenstates to the boundaries and is typically seen in non-Hermitian lattice models. How to predict this, and how to quantify the sensitivity of the spectrum to the boundary conditions are therefore questions that are also studied in this thesis.