  We compare three transitivity properties of finite graphs, namely, for a
positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and
$s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive
but not $(s+1)$-arc transitive then $s\leq 7$ and $s\neq 6$. We show that there
are infinitely many geodesic transitive graphs with this property for each of
these values of $s$, and that these graphs can have arbitrarily large diameter
if and only if $1\leq s\leq 3$. Moreover, for a prime $p$ we prove that there
exists a graph of valency $p$ that is 2-geodesic transitive but not 2-arc
transitive if and only if $p\equiv 1\pmod 4$, and for each such prime there is
a unique graph with this property: it is an antipodal double cover of the
complete graph $K_{p+1}$ and is geodesic transitive with automorphism group
$PSL(2,p)\times Z_2$.
