Recent results on the moment determinacy of probability distributions

We deal with probability distributions, one-dimensional, multi-dimensional or distributions of stochastic processes (e.g. SDEs). One of the assumption is that they have all moments finite. It is well-known from the classical moment problem that any such a distribution is either uniquely determined by its moment (M- determinate) or it is non-unique (M-indeterminate). Along a brief discussion on known conditions (Cramer, Carleman, Krein, Hardy), the emphasis will be on new and checkable conditions for M-determinacy and/or M-indeterminacy of functional transformations of random data. The new results will be reported with hints for their proof. Examples and counterexamples will be given also open questions and conjectures outlined.