  We classify topological phases of non-Hermitian systems in the
Altland-Zirnbauer classes with an additional reflection symmetry in all
dimensions. By mapping the non-Hermitian system into an enlarged Hermitian
Hamiltonian with an enforced chiral symmetry, our topological classification is
thus equivalent to classifying Hermitian systems with both chiral and
reflection symmetries, which effectively change the classifying space and shift
the periodical table of topological phases. According to our classification
tables, we provide concrete examples for all topologically nontrivial
non-Hermitian classes in one dimension and also give explicitly the topological
invariant for each nontrivial example. Our results show that there exist two
kinds of topological invariants composed of either winding numbers or
$\mathbb{Z}_2$ numbers. By studying the corresponding lattice models under the
open boundary condition, we unveil the existence of bulk-edge correspondence
for the one-dimensional topological non-Hermitian systems characterized by
winding numbers, however we did not observe the bulk-edge correspondence for
the $\mathbb{Z}_2$ topological number in our studied $\mathbb{Z}_2$-type model.
