Abstarct: Aabstract In this thesis, we present a Gauss-Jacobi spectral-collocation method with high performance for numerical solving of various types of fractional-delay differential equations. The procedure is that we first replace the delay function in the equation and obtain an equivalent differential system. Then, we use Gauss-Jacobi spectral-collocation method to discretize the gained system in Gauss-Jacobi points and finally we obtain a system of algebraic equations. One of the advantages of this method is to obtain the approximate solution and its fractional derivative, simultaneously. Next, we investigate the convergence analysis of the approximate solution in spaces L1 !;(I) and L2 !;(I) and demonstrate the efficiency and superiority of our method by presenting several numerical examples.