  Exceptional points are universal level degeneracies induced by
non-Hermiticity. Whereas past decades witnessed their new physics, the unified
understanding has yet to be obtained. Here we present the complete
classification of generic topologically stable exceptional points according to
two types of complex-energy gaps and fundamental symmetries of charge
conjugation, parity, and time reversal. This classification reveals unique
non-Hermitian gapless structures with no Hermitian analogs and systematically
predicts unknown non-Hermitian semimetals and nodal superconductors; a
topological dumbbell of exceptional points in three dimensions is constructed
as an illustration. Our work paves the way toward richer phenomena and
functionalities of exceptional points and non-Hermitian topological semimetals.
