Let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}>0$, denote a real $n$-dimensional multisequence of degree $m$. The \textit{Truncated Moment Problem} for $\beta$ (TMP) concerns the existence of a positive Borel measure $\mu$, supported in $\mathbb{R}^{n}$, such that $ \beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m). $ (Here, for $x\equiv (x_{1},\ldots,x_{n})\in \mathbb{R}^{n}$ and $i\equiv (i_{1},\ldots,i_{n})\in \mathbb{Z}_{+}^{n}$, we set $|i| = i_{1}+\cdots + i_{n}$ and $x^{i} = x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$.) A measure $\mu$ as above is a {\it{representing measure}} for $\beta$. We discuss three equivalent ``solutions" to TMP, based on: 1) flat extensions of moment matrices, 2) positive extensions of Riesz functionals, and 3) the \textit{core variety} of a multisequence. In work with G. Blekherman [J. Operator Theory, to appear] We proved that $\beta$ has a representing measure if and only if the core variety is nonempty, in which case the core variety is the union of supports of all finitely atomic representing measures. We discuss open questions concerning difficulties in applying of any of the above solutions to TMP in special cases or in numerical examples.