The theme of this thesis can be divided into two major aspects. The first aspect is to study the delay differential equations and to give some examples of them. The second aspect is to give a new approach for solving delay differential equations (numerically-approximately); this approach depends mainly on Gaussian quadrature numerical integration method and Cubic spine interpolator functions for the unknown exact solution. Also, this approach had been examined successively to the three types of delay differential equations (retarded, neutral and mixed). The suggested approach was applied also to delay differential equation model of the growth of two species of plankton having competitive and allelopathic effects on each other. For each of the discussed types of delay differential equations a computer program is designed in Quick-Basic language and the obtained results are compared with the exact solution or with the previous method (Runge-Kutta method).