The main discussion will be on probability distributions, continuous or discrete, and their determinacy in terms of the moments or in terms of the cumulants. Some results and examples will be presented publicly for the very first time. It will be shown that a few new conditions of modifications of existing conditions can guarantee that a distribution is either M-determinate or M-indeterminate. All conditions involved are easy to be checked. The results will be illustrated by examples. If time, a couple of open questions will be outlined.