In this talk, we will present some new limiting theorems for a family of signed distributions on square-free numbers. These distributions arise naturally from the study of the Moebius function and allowed us to prove "signed" versions of the Erdos-Kac theorem. We will also discuss the intricate connection of the limiting distributions with the generalized Dickman-de Bruijn distributions and dependence of the limiting behavior on the smoothness of the test functions. Joint work with D. Li and Ya. G. Sinai.