The general belief is that attractors of diffeomorphisms of smooth manifolds either have measure zero, or coincide with the phase space. We prove that in the space of diffeomorphisms of a manifold with boundary onto itself there exists an open set (with at most a countable number of hypersurfaces deleted) such that any map from this set has a thick attractor: an attractor that has positive Lebesgue measure together with its complement. The result heavily relies upon the following two: ergodic theorems about the Hausdorff dimension of "exclusive" sets of some particular hyperbolic maps (P.Saltykov; Yu.Ilyashenko); overcoming of the "Fubini nightmare" for some perturbations of partially hyperbolic diffeomorphisms (joint work with A.Negut). Methods developed go back to investigations of A.Gorodetski and the speaker started at late 90's.