On local smoothness of generalized and hypoellipticity of second order differential equations

The paper gives a survey of results on hypoellipticity of second order differential operators. In particular, our new theorems together with earlier results give a necessary and sufficient condition for hypoellipticity of a general second order differential operator with analytic coefficients, provided that it is not fully degenerate at any point of the domain.
We dedicate this paper to I. G. Petrovskii on the occasion of his seventieth birthday.