  We study ultracold color fermions with three internal states Red, Green and
Blue with ${\rm SU(3)}$ symmetry in optical lattices, when color-orbit coupling
and color-flip fields are present. This system corresponds to a generalization
of two-internal state fermions with ${\rm SU(2)}$ symmetry in the presence of
spin-orbit coupling and spin-flipping Zeeman fields. We investigate the
eigenspectrum and Chern numbers to describe different topological phases that
emerge in the phase diagrams of color-orbit coupled fermions in optical
lattices. We obtain the phases as a function of artificial magnetic,
color-orbit and color-flip fields that can be independently controlled. For
fixed artificial magnetic flux ratio, we identify topological quantum phases
and phase transitions in the phase diagrams of chemical potential versus
color-flip fields or color-orbit coupling, where the chirality and number of
midgap edge states changes. The topologically non-trivial phases are classified
in three groups: the first group has total non-zero chirality and exhibit only
the quantum charge Hall effect; the second group has total non-zero chirality
and exhibit both quantum charge and quantum color Hall effects; and the third
group has total zero chirality, but exhibit the quantum color Hall effect.
These phases are generalizations of the quantum Hall and quantum spin Hall
phases for charged spin-$1/2$ fermions. Lastly, we also describe the color
density of states and a staircase structure in the total and color filling
factors versus chemical potential for fixed color-orbit, color-flip and
magnetic flux ratio. We show the existence of incompressible states at rational
filling factors precisely given by a gap-labelling theorem that relates the
filling factors to the magnetic flux ratio and topological quantum numbers.
