This paper is mainly concerned with Nonlinear instability of dispersive waves. lt deals with some advanced studies of instabilities of linear and nonlinear dispersive waves in water of finite and infinite depths. The paper has been organized in the following manner: Section 1 contains the preliminary introduction of the hydrodynamic instability. These preliminaries of hydrodynamic stability are discussed in detail in section 2. The advanced instability studies are presented in the following four sections which are related to linear and nonlinear ocean waves. Section 3 contains the nonlinear instability of water waves due to Benjamin and Feir [1]. The Rayleigh-Taylor linear instability of dispersive waves is illustrated in section 4, where as their nonlinear instability is discussed in section 5. This article concludes with the brief study of Kelvin-Helmholtz linear instability with two inviscid streams which is presented in section 6.