  Finite Unified Theories (FUTs) are N=1 supersymmetric Grand Unified Theories,
which can be made all-loop finite, both in the dimensionless (gauge and Yukawa
couplings) and dimensionful (soft supersymmetry breaking terms) sectors. This
remarkable property, based on the reduction of couplings at the quantum level,
provides a drastic reduction in the number of free parameters, which in turn
leads to an accurate prediction of the top quark mass in the dimensionless
sector, and predictions for the Higgs boson mass and the supersymmetric
spectrum in the dimensionful sector. Here we examine the predictions of two
such FUTs. Next we consider gauge theories defined in higher dimensions, where
the extra dimensions form a fuzzy space (a finite matrix manifold). We
reinterpret these gauge theories as four-dimensional theories with Kaluza-Klein
modes. We then perform a generalized a la Forgacs-Manton dimensional reduction.
We emphasize some striking features emerging such as (i)the appearance of
non-Abelian gauge theories in four dimensions starting from an Abelian gauge
theory in higher dimensions, (ii) the fact that the spontaneous symmetry
breaking of the theory takes place entirely in the extra dimensions and (iii)
the renormalizability of the theory both in higher as well as in four
dimensions. Then reversing the above approach we present a renormalizable four
dimensional SU(N) gauge theory with a suitable multiplet of scalar fields,
which via spontaneous symmetry breaking dynamically develops extra dimensions
in the form of a fuzzy sphere $S^2_N$.
