The talk is devoted to new results in the theory of DP-transformations (transformations preserving the Hausdorff dimension); new results related to the problem of faithfulness resp. non-faithfulness of Vitali coverings generated by many famous expansions of real numbers; new approaches for the study of fractal properties of singularly continuous probability measures.

Special attention will be paid to the interrelations of the theory of DP-transformations and the problem of faithfulness of Vitali coverings.

Applications of obtained results in the Number Theory and in Probability Theory will also be discussed.