In this project, we investigate exact solutions of some nonlinear dispersive wave equations by using different methods such as the tanh method, Jacobi elliptic function method, Sinh-Gordon equation expansion method and Weierstrass elliptic function method. We derive the theta functions by introducing a specific heat conduction problem. Then, we define some Jacobi elliptic functions in relation with the theta functions and we also present some properties and the derivatives of them. Then, we show some ideas about nonlinearity and dispersion of the wave equation. Finally, we get the solitary wave solution of Korteweg-de Vries (KLV)equation. The tanh method has been used to get the solution of the modified Kortewegde Vries (mK-dV) equation and the sinh-Poisson equation with some special cases. We use the Jacobi elliptic function method to derive the solitary wave solution and the triangular periodic solution for some special- type nonlinear equations and coupled of equations. We establish the algorithm of the sinh-Gordon equation expansion method and we deal with some examples with the aid of it. We apply Weierstrass elliptic function method to solve the mKDV equation and the new Hamiltonian amplitude equation. The periodic wave solutions in terms of Jacobi elliptic function solutions and their infinite period counterparts such as solitary wave solutions, shock wave solutions and singular wave solutions have been drawn to illustrate the evolution of the waves in time using arbitrary values of the parameters. The figures have been drawn using Maple.