  We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, that has
bases elements indexed by multiset partitions, where $x$ is an indeterminate
and $r$ and $k$ are non-negative integers. This algebra can be realized as a
diagram algebra that generalizes the partition algebra. When $x$ is an integer
greater or equal to $2r$, we show that ${\rm M\!P}_{r,k}(x)$ is isomorphic to a
centralizer algebra of the symmetric group, $S_n$, acting on the polynomial
ring on the variables $x_{ij}$, $1\leq i \leq n$ and $1\leq j\leq k$. We
describe the representations of ${\rm M\!P}_{r,k}(x)$, branching rule and
restriction of its representations in the case that $x$ is an integer greater
or equal to $2r$.
