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.