Finite SU(N)k unification

We consider N=1 supersymmetric gauge theories based on the group SU(N)1 x SU(N)2 x ... x SU(N)k with matter content (N,N*,1,...,1)+(1,N,N*,...,1)+ ... +(N*,1,1,...,N) as candidates for the unification symmetry of all particles. In particular we examine to which extent such theories can become finite and we find that a necessary condition is that there should be exactly three families. We discuss further some phenomenological issues related to the cases (N,k) = (3,3), (3,4), and (4,3), in an attempt to choose those theories that can become also realistic. Thus we are naturally led to consider the SU(3)3 model which we first promote to an all-loop finite theory and then we study its additional predictions concerning the top quark mass, Higgs mass and supersymmetric spectrum. (author)