In this project, we show how to solve linear and nonlinear ordinary differential equations by using Adomian decomposition method. Linear and nonlinear differential equations arise in all fields of applied mathematics, physical science and engineering, hence being of fundamental importance the existence of methods to find their solutions. The Adomian decomposition method has been applied to a wide class of stochastic and deterministic problems in physics, biology and chemical reactions. For nonlinear models, the method has shown reliable results in supplying analytical approximation that converges very rapidly. However, the implementation of the decomposition method mainly depends upon the computations of Adomian polynomials for nonlinear operators. So developing some practical methods for the computations of Adomian polynomials for all forms of nonlinearity is vital to solve nonlinear problems in many applied sciences. In this project, we present some preliminaries and backgrounds of the method and we explained about the Adomian decomposition method on ODEs. We have given some examples of ordinary differential equations that are computed by Adomian decomposition method. The application of ADM is also one part of this project.