First we recall briefly the classical moment problem and some recent developments. New (and unexpected) results will be presented about important distributions: normal, log-normal, exponential type, etc. Then we perform moment analysis of the distributions of functionals of the Brownian motion (BM): stochastic integrals and stochastic differential equations. We describe explicitly cases when the distributions are unique in terms of the moments and other cases of nonunique distributions. In particular, our favourite geometric BM has moment-nonunique distributions. Related topics, including open questions, will also be discussed.