Recent results on the moment determinacy of probability distributions

We start with two recent probabilistic problems. Then we turn to the main objects of our discussion: heavy tailed probability distributions, onedimensional or multidimensional, with finite all moments. Either such a distribution is uniquely determined by its moments (M-determinate), or it is nonunique (M-indeterminate). Our goal is to describe the current state of art in this area. Some open questions will be outlined. The talk is based on the recent author's works.