The paper addresses the non-Hermitian skin effect (NHSE) in arbitrary dimensions, a phenomenon characterized by the accumulation of eigenstates at system boundaries in non-Hermitian systems. While well-understood in one dimension through non-Bloch band theory, the NHSE in higher dimensions faces significant challenges due to the complexity of open boundary conditions and lattice geometries. The authors present a geometry-adaptive non-Bloch band theory based on spectral potential, which accurately determines the energy spectra, density of states, and generalized Brillouin zone (GBZ) for a given geometry in the thermodynamic limit (TDL). They classify NHSE into critical and non-reciprocal types using net winding numbers. In the critical case, they identify scale-free skin modes on the boundary, while in the non-reciprocal case, various forms of skin modes, including normal or anomalous corner modes, boundary modes, and scale-free modes, are observed. The paper also discusses the instability of non-Bloch spectra under weak perturbations and the non-convergence of spectra in the presence of scale-free modes, linking these issues to the non-exchangeability of the zero-perturbation limit and the TDL. The findings provide a unified non-Bloch band theory that incorporates geometric information, offering a comprehensive understanding of NHSE in arbitrary dimensions.The paper addresses the non-Hermitian skin effect (NHSE) in arbitrary dimensions, a phenomenon characterized by the accumulation of eigenstates at system boundaries in non-Hermitian systems. While well-understood in one dimension through non-Bloch band theory, the NHSE in higher dimensions faces significant challenges due to the complexity of open boundary conditions and lattice geometries. The authors present a geometry-adaptive non-Bloch band theory based on spectral potential, which accurately determines the energy spectra, density of states, and generalized Brillouin zone (GBZ) for a given geometry in the thermodynamic limit (TDL). They classify NHSE into critical and non-reciprocal types using net winding numbers. In the critical case, they identify scale-free skin modes on the boundary, while in the non-reciprocal case, various forms of skin modes, including normal or anomalous corner modes, boundary modes, and scale-free modes, are observed. The paper also discusses the instability of non-Bloch spectra under weak perturbations and the non-convergence of spectra in the presence of scale-free modes, linking these issues to the non-exchangeability of the zero-perturbation limit and the TDL. The findings provide a unified non-Bloch band theory that incorporates geometric information, offering a comprehensive understanding of NHSE in arbitrary dimensions.