  We introduce the exceptional topological insulator (ETI), a non-Hermitian
topological state of matter that features exotic non-Hermitian surface states
which can only exist within the three-dimensional topological bulk embedding.
We show how this phase can evolve from a Weyl semimetal or Hermitian
three-dimensional topological insulator close to criticality when
quasiparticles acquire a finite lifetime. The ETI does not require any symmetry
to be stabilized. It is characterized by a bulk energy point gap, and exhibits
robust surface states that cover the bulk gap as a single sheet of complex
eigenvalues or with a single exceptional point. The ETI can be induced
universally in gapless solid-state systems, thereby setting a paradigm for
non-Hermitian topological matter.
