In this research paper, we applied the Adomian's decomposition method to determine the analytical exact solutions of linear and nonlinear Goursat problems which play very important part in applied and engineering sciences. The proposed technique is fully compatible with the complexity of these problems and obtained results are highly encouraging. Some examples with closed form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient. presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives. The Adomian's Decomposition Method has been shown to solve effectively, easily, and accurately a large class of linear and nonlinear problems, generally two or three iterations lead to high accurate solutions. The basic motivation of the present study is to extend the application of Adomian's decomposition method to linear and nonlinear Goursat problems.