Abstract：In this talk the main discussion will be on probability distributions, 1-dimensional or multi-dimensional, with all moments finite. We analyse easily checkable conditions for M-determinacy or M-indeterminacy and establish useful relationships between them. New conditions and results will be presented for both continuous and discrete distributions. Illustrative examples will be given. If time permits, a few open questions will be outlined.  (Some of the results are based on joint work with GD Lin, P Kopanov, E Yarovaya and K Kostyashin.)