We investigate solutions of the coupled Diral Klein-Gordon Equations in one and three space dimensions. Through analysis of the Fourier representations of the solutions to these equations, we introduce the ‘Null Structure’ as developed by Klainerman and Machedon. This structure allows us to prove the necessary estimates, both fixed time and bilinear space-time, that allow us to show existence of solutions of these equations with initial data of lower regularity than previously required. We also study global existence for a two dimensional wave equation with a critical non-linearity.