  Honeycomb structure has a natural extension to the three dimensions. Simple
examples are hyperhoneycomb and stripy-honeycomb lattices, which are realized
in $\beta $-Li$_{2}$IrO$_{3}$ and $\gamma $-Li$_{2}$IrO$_{3}$, respectively. We
propose a wide class of three-dimensional (3D) honeycomb lattices which are
loop-nodal semimetals. Their edge states have intriguing properties similar to
the two-dimensional honeycomb lattice in spite of dimensional difference.
Partial flat bands emerge at the zigzag or beard edge of the 3D honeycomb
lattice, whose boundary is given by the Fermi loop in the bulk spectrum.
Analytic solutions are explicitly constructed for them. On the other hand,
perfect flat bands emerge in the zigzag-beard edge or when the anisotropy is
large. All these 3D honeycomb lattices become strong topological insulators
with the inclusion of the spin-orbit interaction. Furthermore, point-nodal
semimetals may be realized in the presence of both the antiferromagnetic order
and the spin-orbit interaction.
