Our discussion will be on distributions, continuous or discrete, with finite all moments of positive integer order. Any such a distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). We are going to explain why some distributions are M-determinate, and others are M-indeterminate. The uniqueness, or M-determinacy, is an important property from both theoretical and applied point of view. In particular, the M-determinacy is an essential requirement for the validity of a fundamental limit theorem. We will describe the current state of arts and concentrate on a variety of checkable conditions which are either sufficient or necessary for uniqueness or for non-uniqueness (Cramer, Carleman, Hardy, Krein, Lin, rate of growth of moments, etc.)
Besides the moments, we will exploit the cumulants/semi-invariants to establish a non-conventional limit theorem. We illustrate ideas and results by examples based on distributions such that Normal, Poisson, Exponential, Lognormal. It time permits, some challenging open questions will be outlined.