  Classifications of symmetry-protected topological (SPT) phases provide a
framework to systematically understand the physical properties and potential
applications of topological systems. While such classifications have been
widely explored in the context of Hermitian systems, a complete understanding
of the roles of more general non-Hermitian symmetries and their associated
classification is still lacking. Here, we derive a periodic table for
non-interacting SPTs with general non-Hermitian symmetries. Our analysis
reveals novel non-Hermitian topological classes, while also naturally
incorporating the entire classification of Hermitian systems as a special case
of our scheme. Building on top of these results, we derive two independent
generalizations of Kramers' theorem to the non-Hermitian setting, which
constrain the spectra of the system and lead to new topological invariants. To
elucidate the physics behind the periodic table, we provide explicit examples
of novel non-Hermitian topological invariants, focusing on the symmetry classes
in zero, one and two dimensions with new topological classifications (e.g.
$\mathbb{Z}$ in 0D, $\mathbb{Z}_2$ in 1D, 2D). These results thus provide a
framework for the design and engineering of non-Hermitian symmetry-protected
topological systems.
