  A quadrangular embedding of a graph in a surface $\Sigma$, also known as a
quadrangulation of $\Sigma$, is a cellular embedding in which every face is
bounded by a $4$-cycle. A quadrangulation of $\Sigma$ is minimal if there is no
quadrangular embedding of a (simple) graph of smaller order in $\Sigma$. In
this paper we determine $n(\Sigma)$, the order of a minimal quadrangulation of
a surface $\Sigma$, for all surfaces, both orientable and nonorientable.
Letting $S_0$ denote the sphere and $N_2$ the Klein bottle, we prove that
$n(S_0)=4, n(N_2)=6$, and $n(\Sigma)=\lceil
(5+\sqrt{25-16\chi(\Sigma)})/2\rceil$ for all other surfaces $\Sigma$, where
$\chi(\Sigma)$ is the Euler characteristic. Our proofs use a `diagonal
technique', introduced by Hartsfield in 1994. We explain the general features
of this method.
