  A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex
is uniquely determined by its vector of distances to the vertices in $S$. The
\emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of
$G$. Let $\mathcal{G}_{\beta,D}$ be the set of graphs with metric dimension
$\beta$ and diameter $D$. It is well-known that the minimum order of a graph in
$\mathcal{G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this
paper is to characterise the graphs in $\mathcal{G}_{\beta,D}$ with order
$\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was
previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is
to determine the maximum order of a graph in $\mathcal{G}_{\beta,D}$ for all
values of $D$ and $\beta$. Only a weak upper bound was previously known.
