  A non-complete graph $\Gamma$ is said to be $(G,2)$-distance transitive if
$G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on
the vertex set of $\Gamma$, and for any vertex $u$ of $\Gamma$, the stabilizer
$G_u$ is transitive on the sets of vertices at distance 1 and 2 from $u$. This
paper investigates the family of $(G,2)$-distance transitive graphs that are
not $(G,2)$-arc transitive. Our main result is the classification of such
graphs of valency not greater than 5.
