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 $m$ and let $K$ denote a closed subset of $\mathbb{R}^{n}$. The \textit{Truncated $K$-Moment Problem} concerns the existence of a \textit{$K$-representing measure} for $\beta$, i.e., a positive Borel measure $\mu$, supported in $K$, such that \begin{equation} \beta_{i} = \int_{K} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m). \end{equation} Let $\mathcal{P}_{m} := \{p\in \mathbb{R}[x_{1},\ldots,x_{n}]: ~~deg~p\le m\}$. We associate to $\beta$ the \textit{Riesz functional} $L_{\beta}:\mathcal{P}_{m} \mapsto \mathbb{R}$ defined by $L_{\beta}(\sum a_{i}x^{i}) = \sum a_{i}\beta_{i}$. The existence of a $K$-representing measure implies that $L_{\beta}$ is \textit{$K$-positive}, i.e., if $p\in \mathcal{P}_{m}$ satisfies $p|K\ge 0$, then $L_{\beta}(p)\ge 0$. In the \textit{Full $K$-Moment Problem} for $\beta \equiv \beta^{(\infty)}$, a classical theorem of M. Riesz ($n=1$) and E.K. Haviland $(n>1$) shows that $\beta$ has a $K$-representing measure if and only if $L_{\beta}$ is $K$-positive. In the Truncated $K$-Moment Problem, the direct analogue of Riesz-Haviland is not true. We discuss the gap between $K$-positivity and the existence of $K$-representing measures, with reference to Tchakaloff's Theorem, approximate $K$-representing measures, a ``truncated" Riesz-Haviland theorem due to Curto-F., a ``strict" K-positivity existence theorem of F.-Nie, and recent results concerning the \textit{core variety} of a multisequence.