  Non-Hermitian Hamiltonians, which describe a wide range of dissipative
systems, and higher-order topological phases, which exhibit novel boundary
states on corners and hinges, comprise two areas of intense current research.
Here we investigate systems where these frontiers merge and formulate a
generalized biorthogonal bulk-boundary correspondence, which dictates the
appearance of boundary modes at parameter values that are, in general,
radically different from those that mark phase transitions in periodic systems.
By analyzing the interplay between corner/hinge, edge/surface and bulk degrees
of freedom we establish that the non-Hermitian extensions of higher-order
topological phases exhibit an even richer phenomenology than their Hermitian
counterparts and that this can be understood in a unifying way within our
biorthogonal framework. Saliently this works in the presence of the
non-Hermitian skin effect, and also naturally encompasses genuinely
non-Hermitian phenomena in the absence thereof.
