Colli-Vargas' conjecture. Every two-dimensional diffeomorphism with homoclinic tangency can be approximated in the Cr-topology by diffeomorphisms
having non-trivial wandering domains.

Newhouse showed that there is an open set of two-dimensional diffeomorphisms in Diffr(M) (r >1) having homoclinic tangencies. Such an open set is
called a Cr Newhouse open set. Hence, the above conjecture can be rephrased as "any Cr Newhouse open set contains a dense subset of diffeomorphisms with non-trivial wandering domains". We show that it is true, and moreover present an affirmative answer to Takens last problem.