  In this article we prove two non-existence results for translating solitons
of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We
also obtain an upper bound to the maximum height that a compact embedded
translator in $\mathbb{R}^{3}$ can achieve. On the other hand, we study
graphical perturbations of translators, showing that asymptotic graphical
perturbations of a graph translator of revolution remain a hypersurface of
revolution. Finally, we prove that compact translators that lie between two
parallel planes inherit the symmetries of their boundaries curves.
