  Let $\mathcal M_{mn}=\mathcal M_{mn}(\mathbb F)$ denote the set of all
$m\times n$ matrices over a field $\mathbb F$, and fix some $n\times m$ matrix
$A\in\mathcal M_{nm}$. An associative operation $\star$ may be defined on
$\mathcal M_{mn}$ by $X\star Y=XAY$ for all $X,Y\in\mathcal M_{mn}$, and the
resulting \emph{sandwich semigroup} is denoted $\mathcal M_{mn}^A=\mathcal
M_{mn}^A(\mathbb F)$. These semigroups are closely related to Munn rings, which
are fundamental tools in the representation theory of finite semigroups. In
this article, we study $\mathcal M_{mn}^A$ as well as its subsemigroups
$\operatorname{Reg}(\mathcal M_{mn}^A)$ and $\mathcal E_{mn}^A$ (consisting of
all regular elements and products of idempotents, respectively), as well as the
ideals of $\operatorname{Reg}(\mathcal M_{mn}^A)$. Among other results, we:
characterise the regular elements, determine Green's relations and preorders,
calculate the minimal number of matrices (or idempotent matrices, if
applicable) required to generate each semigroup we consider, and classify the
isomorphisms between finite sandwich semigroups $\mathcal M_{mn}^A(\mathbb
F_1)$ and $\mathcal M_{kl}^B(\mathbb F_2)$. Along the way, we develop a general
theory of sandwich semigroups in a suitably defined class of \emph{partial
semigroups} related to Ehresmann-style "arrows only" categories, we hope this
framework will be useful in studies of sandwich semigroups in other categories.
We note that all our results have applications to the \emph{variants} $\mathcal
M_n^A$ of the full linear monoid $\mathcal M_n$ (in the case $m=n$), and to
certain semigroups of linear transformations of restricted range or kernel (in
the case that $\operatorname{rank}(A)$ is equal to one of $m,n$).
