  We study the dependence on the spatial dimensionality of different quantities
relevant in the description of the Anderson transition by combining numerical
calculations in a $3 \leq d \leq 6$ disordered tight binding model with
theoretical arguments. Our results indicate that, in agreement with the one
parameter scaling theory, the upper critical dimension for localization is
infinity. Typical properties of the spectral correlations at the Anderson
transition such as level repulsion or a linear number variance are still
present in higher dimensions though eigenvalues correlations get weaker as the
dimensionality of the space increases. It is argued that such a critical
behavior can be traced back to the exponential decay of the two-level
correlation function in a certain range of eigenvalue separations. We also
discuss to what extent different effective random matrix models proposed in the
literature to describe the Anderson transition provide an accurate picture of
this phenomenon. Finally, we study the effect of a random flux in our results.
