TeV scale models with two supersymmetries

The construction of models of low energy physics with two supersymmetries is discussed. All such SU(3) x SU(2) x U(1) models appear to be beset by insurmountable problems. A SU(3) x SU(4) x Y(1) model is constructed which gives a natural distinction between mirror fermions and ordinary fermions. Many N=1 supersymmetric models describing particle physics at the TeV scale have been written down. These theories have more particles than the standard model. However they are constrained in new ways, and are less divergent than the standard model. They offer explanations, to varying degrees, of the gauge hierarchy problem, and many would be relieved and excited by the observation of a superpartner particle. In this talk the authors wish to go one step further and discuss theoretical prejudices for and against the possibility that experimentalists will soon observe two superpartners for each known particle. Models with two supersymmetries have greatly improved convergence properties. Indeed, theories considered here could be the low energy remnants of a finite grand unified theory. These models have several new constraints. For example, all dimension four interactions are governed by gauge couplings. Constructing fermion mass matrices becomes quite a challenge