  We give (necessary and sufficient) conditions over a sequence $\left\{
f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized
Stieltjes moment problem \[ \int_{0}^{\infty} f_{n}(x)\phi(x)\mathrm{d}
x=a_{n}, \ \ \ n\in\mathbb{N}, \] has solutions
$\phi\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp}
\phi\subseteq[0,\infty)$. Furthermore, we consider more general problems of
this kind for measure or distribution sequences $\left\{ f_{n}\right\}
_{n=0}^{\infty}$. We also study vector moment problems with values in Frechet
spaces and multidimensional moment problems.
