In chapter 1, starting with linear and nonlinear waves, we introduce the concept of solitary wave solutions (SWS) which arises due to a balance in weak nonlinearity and dispersion and compacton solutions which are a result of strong nonlinearity and dispersion. In chapter 2, SWSs of Korteweg de Vries (KJV) equation, a modified KdV (mKdV) equation and Boussinesq equation have been derived. These equa tions are models for nonlinear long waves in shallow water and several other physical situations. The generalized KDV equation known as Kadomtsev-Petviashvili (KP) equation in two space variables has also been solved using the Adomian Decompo sition method to derive SWSs. In chapter 3, Tanh-coth method has been used to derive SWSs of KDV equation , mKdV equation, a generalized KdV equation in one variable, a generalized KdV equation with two power nonlinearities, the potential equation and the Gardner equation. In chapter 4, compacton solutions have been derived for K(n,n) equation and K(n+1,n+1) equations. Maple has been used to draw graphs of many solutions.