Geometrical study of holomorphic foliations of the complex plane, both projective and affine, lies on the boundary of differential equations, topology and complex analysis. Foliations of \Bbb CP^2 have an algebraic origin: they are defined by polynomial vector fields, but their behavior is highly transcendental. Their properties are drastically different from those of real polynomial vector fields. Properties of density of leaves, absolute rigidity and existence of a countable number of limit cycles were discovered by different authors in 60s and 70s. The talk will present these results together with a survey of the further development and open problems.

Foliations of \Bbb C^2 have an analytic origin: they are defined by analytic vector fields. Generic properties of these fields were studied very recently. Yet genericity of density of leaves and existence of the infinite number of complex limit cycles is recently proved. Moreover, generic leaves of such foliations are either disks, or cylinders. These results are obtained by graduate students Firsova, Kutuzova and Volk.