  The subdivision graph $S(\Sigma)$ of a graph $\Sigma$ is obtained from
$\Sigma$ by `adding a vertex' in the middle of every edge of $\Si$. Various
symmetry properties of $\S(\Sigma)$ are studied. We prove that, for a connected
graph $\Sigma$, $S(\Sigma)$ is locally $s$-arc transitive if and only if
$\Sigma$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of
$S(\Sigma)$ is $2d+\delta$, where $\Sigma$ has diameter $d$ and $0\leqslant
\delta\leqslant 2$, and local $s$-distance transitivity of $\S(\Sigma)$ is
defined for $1\leqslant s\leqslant 2d+\delta$. In the general case where
$s\leqslant 2d-1$ we prove that $S(\Sigma)$ is locally $s$-distance transitive
if and only if $\Sigma$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. For the
remaining values of $s$, namely $2d\leqslant s\leqslant 2d+\delta$, we classify
the graphs $\Sigma$ for which $S(\Sigma)$ is locally $s$-distance transitive in
the cases, $s\leqslant 5$ and $s\geqslant 15+\delta$. The cases $\max\{2d,
6\}\leqslant s\leqslant \min\{2d+\delta, 14+\delta\}$ remain open.
