The search for topological states in non-Hermitian systems, and more specifically in non-Hermitian lattice models, has become a newly emerging research front. Non-Hermitian systems are much more than a theoretical curiosity, they arise naturally in the description of the finite lifetime due to interactions, or more prominently, in photonic or acoustic systems.
What is new about them? What are the key challenges in this field? For a brief summary of the latest research in this area we refer the reader to Refs. 
and . Here we will just mention one of the most striking surprises that we found along our way: A pristine non-Hermitian lattice may have all the states localized at the boundary. This is, the pristine system, no matter how large, becomes devoid of extended states. We reported this finding in Ref.  [submitted on 25Aug2017] where it was connected to higher-order exceptional points and later on called "non-hermitian skin effect" in a later paper by other authors  (published in August 2018).
Exceptional points of an order scaling with the system size (higher-order exceptional points) can, therefore, condense all the states of a system. Like a peculiar kind of Aleph (in the short story The Aleph by J. L. Borges, an Aleph “is one of the points in space that contains all other points”), these points may condense all the eigenstates of a system. But unlike the Aleph, these points destroy information instead of compressing it. It is thus a non-hermitian condensation.
This surprising localization due to the non-hermitian skin effect has been verified in different recent experiments, see  for a recent overview of this subject. The experiments include the realization in non-reciprocal mechanical systems , a quantum walk with photons  and topoelectrical circuits .
You might also enjoy this recent talk giving a brief general overview.
 "Perspective on topological states of non-Hermitian lattices"
L. E. F. Foa Torres
Journal of Physics: Materials 3, 014002 (2020), doi:10.1088/2515-7639/ab4092
 "Non-Hermitian robust edge states in one-dimension:
Anomalous localization and eigenspace condensation at exceptional points"
V. M. Martinez-Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres
Physical Review B 97, 121401(R) (2018)
publisher, full text
 "Edge States and Topological Invariants of Non-Hermitian Systems"
S. Yao and Z. Wang
Phys. Rev. Lett. 121, 086803 (2018)
 "Non-reciprocal robotic metamaterials"
M. Brandenbourger,X. Locsin, and C. Coulais
Nature Communications 10, 4608 (2019)
 "Non-Hermitian bulk–boundary correspondence in quantum dynamics"
L. Xiao et al.
Nature Physics (2020)
 "Reciprocal skin effect and its realization in a topolectrical circuit"
T. Hofmann et al.
Phys. Rev. Research 2, 023265 (2020)