Many of the physical phenomena modeled into nonlinear evolution equations and the special characteristics of the compact finite difference method such as stability, efficiency and high order convergence in time and space, iired us to numerically solve, in this thesis, some well-known nonlinear evolution equations using compact finite difference methods. This thesis can be divided into two main parts: 1. In the first part, we give a short demonstration of evolutionary equations and an introduction to the creation of these types of differential equations. In the past, partial differential equations depending on time were call evolutionary equations, however, nowadays evolution equation is referred to a differentiation rule in a time interval. In the sequence, the compact finite difference method and its construction are briefly discussed. Also we define the finite difference method that can be divided into group: implicit method and explicit method. Compact finite difference method is an implicit method with high accuracy, in which a less number of mesh points is needed in comparison to the explicit method. Also it is observed that the Douglas method can be employed as the first compact finite difference method . In the second part, some types of evolution equations such as Burgers' equation, one dimensional Schr?dinger equation, two dimensional Schr?dinger equation, Generalized Regularized Long Waves ( GRLW ) equation and Korteweg-de Vries ( KdV ) equation are considered and by using the compact finite difference method these equations are numerically solved. The main problem with nonlinear evolution equations is the existence of nonlinear terms in them. Some these equations can be transformed to linear equations. For instance, the Burgers equation can be transformed into a linear equation using Hopf-Cole transformation. However, until now, the number of evolutionary equations transformed into linear ones is not many. For this cause, a method based on the Taylor expansion is presented to enable us in the linearization of the nonlinear term. This linearization trick is used for the Burgers', GRLW and KdV equations. The one dimensional Schr?dinger equation is discreted to a linear equation using central differences. Alternating direction implicit ( ADI ) compact finite difference schemes are devised for the numerical solution of two dimensional Schr?dinger equation. Moreover, the stability and convergence of this method is investigated and a convergence order of O( k^2 + h^4 ) is achieved where " k " and " h " are mesh sizes of time and space variables, respectively. By using a variety of examples and by comparing this method with other methods, the accuracy of this approach is tested and the observation is that the compact finite difference method has a higher accuracy in comparison with the other counterpart methods.