  We deal with solitons of the mean curvature flow. The definition of
\textit{translating solitons on a light-like direction} in Minkowski 3-space is
introduced. Firstly, we classify those which are graphical, \textit{translation
surfaces}, obtaining space-like and time-like, entire and not entire, complete
and incomplete examples. Among them, all our time-like examples are incomplete.
The second family consists of those which are invariant by a 1-dimensional
subgroup of parabolic motions, i.e, with light-like axis. The classification
result implies that all examples of this second family have singularities.
