  The past decades have witnessed an explosion of interest in topological
materials, and a lot of mathematical concepts have been introduced in condensed
matter physics. Among them, the bulk-boundary correspondence is the central
topic in topological physics, which has inspired researchers to focus on
boundary physics. Recently, the concepts of topological phases have been
extended to non-Hermitian Hamiltonians, whose eigenvalues can be complex.
Besides the topology, non-Hermiticity can also cause a boundary phenomenon
called the non-Hermitian skin effect, which is an extreme sensitivity of the
spectrum to the boundary condition. In this article, we review developments in
non-Hermitian topological physics by focusing mainly on the boundary problem.
As well as the competition between non-Hermitian and topological boundary
phenomena, we discuss the topological nature inherent in non-Hermiticity
itself.
