  In this paper we introduce a perturbatively super-renormalizable and unitary
theory of quantum gravity in any dimension D. The theory presents two entire
functions, a.k.a. "form factors", and a finite number of local operators
required by the quantum consistency of the theory itself. The main reason to
introduce the entire functions is to avoid ghosts (states of negative norm)
like the one in the four-dimensional Stelle's theory. The new theory is indeed
ghost-free since the two entire functions have the property to generalize the
Einstein-Hilbert action without introducing new poles in the propagator. The
theory is renormalizable at one loop and finite from two loops upward. In this
paper we essentially study three classes of form factors, systematically
showing the tree-level unitarity. We prove that the gravitation potential is
regular in r = 0 for all the choices of form factors compatible with
renormalizability and unitarity. We also include Black hole spherical symmetric
solutions omitting higher curvature corrections to the equation of motions. For
two out of three form factors the solutions are regular and the classical
singularity is replaced by a "de Sitter-like core" in r=0. For one particular
choice of the form factors, we prove that the D-dimensional "Newtonian
cosmology" is singularity-free and the Universe spontaneously follows a de
Sitter evolution at the "Planck scale" for any matter content. We conclude the
article stating that, in the ultraviolet regime, the spectral dimension takes
on different values for the three cases: less than or equal to "1" for the
first case, "0" for the second one and "2" for the third one. Once the class of
theories compatible with renormalizability and unitarity is defined, the
spectral dimension has the same short-distance "critical value" or
"accumulation point" for any value of the topological dimension D.
