In this thesis, we investigate exact analytical solutions of certain nonlinear disper sive wave equations by using different methods such as mapping methods through Jacobi elliptic functions (JEFs), tanh-method and G"/G expansion method. We also use Adomian decomposition method (ADM) to find approximate solutions whose convergence to the exact solutions has been investigated. In chapter 2, we use the mapping methods to solve a coupled wave equation which governs shallow water waves for two-layered fluid flow. The solitary wave solutions (SWSs), kink wave and singular wave solutions have been derived when the modulus m of the JĘFs is close to 1. Using ADM, approximate solutions have been derived and the convergence to the exact solutions has been established. In chapter 3, the same coupled wave equation and the coupled Klein-Gordon equation which is a model equation in quantum field theory, have been solved using Go/G expansion method. Kink wave and singular wave solutions for certain values of parameters have been deduced. In chapter 4, nonlinear schrodinger equation (NLSE) in metamaterials has been solved using mapping methods. In chapter 5, a generalized fractional Boussinesq equation has been solved using mapping methods. Correctness of the solutions and the constraint conditions for the NLSE and Boussinesq equation have been confirmed by tanh-method.