This paper can be considered as the end of a long journey started by \textit{W. T. Tutte} [Proc. Camb. Philos. Soc. 43, 459--474 (1947; Zbl 0029.42401)] nearly 70 years ago. This then was continued by \textit{D. M. Goldschmidt} [Ann. Math. (2) 111, 377--406 (1980; Zbl 0475.05043)] and \textit{R. Weiss} [Combinatorica 1, 309--311 (1981; Zbl 0486.05032)] about 35 years ago. Goldschmidt's paper started the amalgam method which became an important tool in the classification of the finite simple groups. Among many papers the maybe most important one the classification of the weak \(BN\)-pairs of rank two due to \textit{A. Delgado} and the second author [``Weak \((B,N)\)-pairs of rank 2'', in: Groups and graphs: new results and methods. Basel-Boston-Stuttgart: Birkhäuser Verlag. 58--244 (1985)] should be mentioned. In the paper under review, the authors study graphs \(\Gamma\) (undirected, without loops and multiple edges) in which any vertex possesses at least three neighbors and groups \(G \leq\Aut(\Gamma)\), which are locally \(s\)-transitive, i.e. for any vertex \(x\), \(|G_x| \leq \infty\) and \(G_x\) acts transitively on all path of length \(r \leq s\) starting at \(x\). The main result is that either \( s \leq 5\) or \(G\) is a group with a weak \(BN\)-pair of rank two, with respect to \(G_x\), \(G_y\), where \(\{x,y\}\) is an edge. In particular, using the paper mentioned above, this shows \(s \leq 9\) in all cases and even \(s \leq 7\) if \(G\) acts vertex transitively. If \(s \geq 2\), then \(G_x\) induces an 2-transitive group on the neighbors \(\Gamma (x)\) of \(x\). Using the classification of the 2-transitive groups (which in turn uses the classification of the finite simple groups), the structure of \(G_x/G_{\Gamma (x)}\) is determined. Now, there is a subdivision in the two cases that \(G_x\) is of local characteristic \(p\) for some prime \(p\) or \(G_x\) has a component. The main step in the proof is to show that if \(s \geq 6\), then there is a prime such that \(G_x\) is of local characteristic \(p\), i.e. \(C_{G_x}(O_p(G_x)) \leq O_p(G_x)\). This is basically a so called theorem of Thompson-Wielandt type. Then, if \(s \geq 6\) methods coming from the amalgam method and which are similar to some approaches in the classification of the finite simple groups give that \(G_x\), \(G_y\), \(\{x,y\}\) an edge, form a weak \(BN\)-pair of rank two. By the result of Delgado and Stellmacher [loc. cit.] we then have that the structure of the groups \(G_x\), \(G_y\) is known.