I introduce a perturbatively super-renormalizable and 
unitary theory of quantum gravity in any dimension D
starting from the four dimensional case. The theory 
presents one entire function, a.k.a. "form factor", and a  
finite number of local operators required by the quantum 
consistency as well as unitarity of the theory itself.
The theory is power-counting renormalizable at one loop 
and finite from two loops upward.
I essentially present three classes of form factors, 
systematically showing the tree-level unitarity. It is right 
now under investigation a possible N=1 supersymmetric  
extension of the theory in four dimensions. Preliminary 
results indicate that the nonlocal supergravity theory is  
power-counting super-renormalizable and tree level 
unitary with the same particle content of the local N=1  
supergravity. In contrast to the local (quadratic-)higher 
derivative supergravity in its nonlocal generalization all 
the states fill up in N=1 supergravity multiplet. We 
believe that the extended  SO(N) supergravity, for N=4 
and/or N=8, can be off-shell divergence-free also at one 
loop. At semiclassical level the gravitational potential is 
regular in r = 0 for all the choices of form factors
compatible with renormalizability and unitarity. For two 
out of three form factors the black hole solutions are
regular and the classical singularity is replaced by a "de 
Sitter-like  core" in r=0.
For one particular example of form factor, I 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.
I conclude 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.