  A non-Hermitian system can exhibit extensive sensitivity of its complex
energy spectrum to the imposed boundary conditions, which is beyond any known
phenomenon from Hermitian systems. In addition to topologically protected
boundary modes, macroscopically many ``skin'' boundary modes may appear under
open boundary conditions. We rigorously derive universal results for
characterizing all avenues of boundary modes in non-Hermitian systems for
arbitrary hopping range. For skin modes, we introduce how exact energies and
decay lengths can be obtained by threading an imaginary flux. Furthermore, for
1D topological boundary modes, we derive a new generic criterion for their
existence in non-Hermitian systems which, in contrast to previous formulations,
does not require specific tailoring to the system at hand. Our approach is
intimately based on the complex analytical properties of in-gap exceptional
points, and gives a lower bound for the winding number related to the vorticity
of the energy Riemann surface. It also reveals that the topologically
nontrivial phase is partitioned into subregimes where the boundary mode's decay
length depends differently on complex momenta roots.
