This paper aims to classify all the traveling
fronts of a curvature flow with external force fields in the
two-dimensional Euclidean space, i.e., the curve is evolved by the
sum of the curvature
and an external force field. We show that any traveling front
is either a line or Grim Reaper if the external force field is constant. However, we find
traveling fronts are of completely different geometry for
non-constant external force fields.