For a positive integer s less than or equal to the diameter of a graph Γ , an s -geodesic of Γ is a path ( v 0 , v 1 , … , v s ) such that the distance between v 0 and v s is s . The graph Γ is said to be s -geodesic transitive, if Γ contains an s -geodesic and its automorphism group is transitive on the set of t -geodesics for all t ≤ s . In particular, if Γ is s -geodesic transitive with s equal to the diameter of Γ , then Γ is called geodesic transitive. In this paper, we classify the family of finite 2-geodesic transitive graphs of valency 6. Then we completely determine such graphs which are geodesic transitive.