Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on $\mathbb C^2$

This paper is a survey of the present state of the problems related to the generic properties of foliations defined on $\mathbb C^2$ by algebraic differential equations. We prove that the properties of density, absolute rigidity, and existence of a countable set of complex limit cycles are inherent in all equations except possibly for the union of some real algebraic set and real analytic set of codimension at least two in the space of coefficients.