  We hereby present a class of multidimensional higher derivative theories of
gravity that realizes an ultraviolet completion of Einstein general relativity.
This class is marked by a "non-polynomal" entire function (form factor), which
averts extra degrees of freedom (including ghosts) and improves the high energy
behavior of the loop amplitudes. By power counting arguments, it is proved that
the theory is super-renormalizable in any dimension, i.e. only one-loop
divergences survive. Furthermore, in odd dimensions there are no counter terms
for pure gravity and the theory turns out to be "finite." Finally, considering
the infinite tower of massive states coming from dimensional reduction, quantum
gravity is finite in even dimension as well.
