We deal with the classical moment problem for probability distributions, one- or multi-dimensional, with finite all moments. Either such a distribution is uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). The well-known classical conditions by Cramer, Carleman and Krein play an essential role, however the emphasis will be on their converses and on new developments obtained over the last two decades. The following specific topics will be presented in detail:
(a) New Hardy’s criterion for uniqueness.
(b) Criteria based on the rate of growth of moments.
(c) Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.
(d) Multidimensional moment problem.
(e) Nonlinear transformations of random data and their moment (in)determinacy.
(f) M-determinacy of distributions of stochastic processes defined by SDEs.
There will be new and well-referenced results, hints for their proof, and illustrations by examples and counterexamples. Some facts may look striking and even shocking. Intriguing open questions and conjectures will be outlined.