  We review topological properties of two series of ternary compounds AMgBi
(A=K, RB, Cs) and ABC with a hexagonal ZrBeSi type structure. The first series
of materials AMgBi are predicted to be topological critical Dirac semimetals.
The second series of ternary compounds, such as KZnP, BaAgAs, NaAuTe and KHgSb,
can be used to realize various topological insulating states and semimetal
states. The states are highly tunable as the realization of these topological
states depends on the competition between several energy scales, including the
energy of atomic orbitals, the energy of crystal splitting, the energy
difference between bonding and antibonding states, and the strength of
spin-orbit coupling. The exotic surface states in these series of compounds are
predicted and are closely related to their unique crystal structures.
