  We consider {translators} (i.e., initial condition of translating solitons)
to mean curvature flow (MCF) in the hyperbolic $3$-space $\mathbb H^3$,
providing existence and classification results. More specifically, we show the
existence and uniqueness of two distinct one-parameter families of complete
rotational translators in $\mathbb H^3$, one containing catenoid-type
translators, and the other parabolic cylindrical ones. We establish a tangency
principle for translators in $\mathbb H^3$ and apply it to prove that properly
immersed translators to MCF in $\mathbb H^3$ are not cylindrically bounded. As
a further application of the tangency principle, we prove that any horoconvex
translator which is complete or transversal to the $x_3$-axis is necessarily an
open set of a horizontal horosphere. In addition, we classify all translators
in $\mathbb H^3$ which have constant mean curvature. We also consider rotators
(i.e., initial condition of rotating solitons) to MCF in $\mathbb H^3$ and,
after classifying the rotators of constant mean curvature, we show that there
exists a one-parameter family of complete rotators which are all helicoidal,
bringing to the hyperbolic context a distinguished result by Halldorsson, set
in $\mathbb R^3$.
