  Chevrel phase materials form a family of ternary molybdenum chalcogenides
with a general chemical formula $A_x{\rm Mo}_6X_8$ ($A$ = metal elements, $X$ =
chalcogen). The variety of $A$ atoms makes a large number of family members and
leads to many tunable physical properties, such as the superconductivity,
thermoelectricity and the ionic conductivity. In this work, we have further
found various nontrivial band topological states in these materials by using
first-principle calculations. The compounds having time-reversal symmetry, such
as ${\rm BaMo}_6{\rm S}_8$, ${\rm SrMo}_6{\rm S}_8$, and ${\rm Mo}_6{\rm S}_8$,
are topological insulators in both of the $R\bar{3}$ and $P\bar{1}$ phases,
whereas ${\rm EuMo}_6{\rm S}_8$ within ferromagnetic state, it is an axion
insulator in the $R\bar{3}$ phase and a trivial one in the $P\bar{1}$ phase.
This indicates that the change of $A$ ions can modify the chemical potential,
lattice distortion, and magnetic orders, which offers a unique way to influence
the topological states and other properties. We hope this work can stimulate
further studies of Chevrel phase materials to find more intriguing phenomena,
such as topological superconducting states and Majorana modes.
