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According to the idea of the SU(7) grand unified theories, an SU(8) grand unified model accommodating three generations of light fermions with V-A coupling for the weak interactions is proposed. The Higgs mechanism of spontaneous breaking and the fermion mass matrices in this model are presented.

Two SU(8) GUT models (one with fractionally charged color singlet fermions, another with low mass magnetic monopoles) are proposed. Both of them can accomodate three generations of ordinary light fermions.

The conditions of finite N=1 supersymmetric theories are discussed in general cases. A large class of N=1 supersymmetric Yang-Mills theories of representations of classical groups finite to all orders is obtained.

In this short note we list possible representations of the classical semisimple groups G 1 × G 2 for finite N = 2 supersymmetric theories.

A two-loop finite SU(6) model is discussed which contains three generations of light quarks and leptons. The masses of quarks and leptons in the model are rather arbitrary except for certain bounds on them. Also the K-M matrix is arbitrary so that the model seems to be a realistic one. The gauge symmetry breaking through SU(6) → SU(3) × SU(2) × U(1) → SU(3) × U(1) em is discussed.

Some SO(18) GUT models are proposed which can accomodate fractionally charged color singlets with ordinary charge ±e 2 and a monopole with ordinary magnetic charge 1 2e . The models are consistent with the Dirac quantisation rule and peculiar photons appear. Future experimental tests of the peculiar photons are discussed. The models can also accomodate four or six generations of ordinary quarks and leptons.

Here, a large class of finite N = 1 supersymmetric Yang-Mills theories of representations of SU( n ) is obtained. There are a few candidates which can accommodate three generations of fermions.

From the requirements of asymptotic freedom, the absence of the triangle anomaly in D = 4 dimensions, the absence of the hexagon anomaly in D = 10 dimensions, and the “generalized Survival Hypothesis”, we find that it is not possible to construct SU(N) GUT models with four generations. But it is possible to build models with three generations, and, there are only two possibilities, namely, SU (8) and SU (9).