We consider functional transformations of random data and describe cases of uniquness of the distributions in terms of the moments and other cases when the distributions are nonunique. We are going to comment on classical results (Carleman, Hausdorff, Cramer, Krein) as well as on some quite recent developments, e.g. how to write Stieltjes classes for M-indeterminate distributions. If time permits, some open questions will be outlined. The speaker will address his talk not only to professionals in mathematics/probability/statistics, but also to PhD students in these areas.