Astrophysical systems have many kinds of hydrodynamic and hydromagnetic instabilities, which are important in understanding their equilibrium configurations and activities. In this and subsequent two chapters we focus our attention to hydrodynamic instabilities. Hydromagnetic instabilities are discussed in part II. Hydrodynamical instabilities are further classified into two classes. One is the instabilities which occur in the framework of non-dissipative processes, and is called dynamical instabilities. Instabilities of the other class are those which are realized by presence of dissipative processes. In this chapter we treat the former dynamical instabilities in the framework of linear analyses. Instabilities due to dissipative processes are considered in Chaps. 7 and 8 .