  In this paper we present a thorough study of transport, spectral and
wave-function properties at the Anderson localization critical point in spatial
dimensions $d = 3$, $4$, $5$, $6$. Our aim is to analyze the dimensional
dependence and to asses the role of the $d\rightarrow \infty$ limit provided by
Bethe lattices and tree-like structures. Our results strongly suggest that the
upper critical dimension of Anderson localization is infinite. Furthermore, we
find that the $d_U=\infty$ is a much better starting point compared to $d_L=2$
to describe even three dimensional systems. We find that critical properties
and finite size scaling behavior approach by increasing $d$ the ones found for
Bethe lattices: the critical state becomes an insulator characterized by
Poisson statistics and corrections to the thermodynamics limit become
logarithmic in $N$. In the conclusion, we present physical consequences of our
results, propose connections with the non-ergodic delocalised phase suggested
for the Anderson model on infinite dimensional lattices and discuss
perspectives for future research studies.
