An s-are in a graph is a vertex sequence (alpha (0), alpha (1),...,alpha (s)) such that (alpha (i-1)alpha (i)) epsilon ET for 1 less than or equal to i less than or equal to s and alpha (i-1) not equal alpha (i+1) for 1 less than or equal to i s less than or equal to - 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + l)-arcs. It is proved that if Gamma is a finite connected s-transitive graph (where s greater than or equal to 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Gamma is a normal cover of the complete bipartite graph K-2m,(2m), or S = 2 and Gamma is a normal cover of one of the following 2-transitive graphs: Kpm+l (the complete graph of order p(m+1)), K-2m,(2m) - 2(m)K(2) (the complete bipartite graph of order 2(m+1) minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.).