**Abstract**

The first part of the contributed chapter discuss the overview of electric propulsion technology and its requirement in different space missions. The technical terms specific impulse and thrust are explained with their relation to exhaust velocity. The shortcoming of the Hall thrusters and its erosion problems of the channel walls are also conveyed. The second part of the chapter discuss the various waves and electromagnetic instabilities propagating in a Hall thruster magnetized plasma. The dispersion relation for the azimuthal growing waves is derived analytically with the help of magnetohydrodynamics theory. It is depicted that the growth rate of the instability increases with magnetic field, electron drift velocity and collisional frequency, whereas it is decreases with the initial drift of the ions.

**Keywords:** electric propulsion, Hall thruster, dispersion, impulse, exhaust velocity, growth rate
