  We consider univariate distributions with finite moments of all positive
orders. The moment problem is to determine whether or not a given distribution
is uniquely determined by the sequence of its moments. There is a huge
literature on this classical topic. In this survey, we will focus only on the
recent developments on the checkable moment-(in)determinacy criteria including
Cramer's condition, Carleman's condition, Hardy's condition, Krein's condition
and the growth rate of moments, which help us solve the problem more easily.
Both Hamburger and Stieltjes cases are investigated. The former is concerned
with distributions on the whole real line, while the latter deals only with
distributions on the right half-line. Some new results or new simple (direct)
proofs of previous criteria are provided. Finally, we review the most recent
moment problem for products of independent random variables with different
distributions, which occur naturally in stochastic modelling of complex random
phenomena.
