Abstract: Stability of an aluminum reduction cell is one of the key technologies in the aluminum electrolyte industry. This paper reviews its historical development, current status and new advances in the past two decades. The physical phenomenon of reduction cell stability and the mechanism of magneto-hydrodynamics stability are explained. The mathematical definition and criteria of stability are given. This paper also presents 2D and 3D mathematical models and numerical methods for stability analysis (finite element method and spectral method). Finally, a stability analysis of a 230KA reduction cell is carried out in this paper.