In this thesis we deal with combinatorial and geometric properties of arc complexes and triangulation graphs, and we will provide some applications to the study of the mapping class group and to the Teichmüller theory of a bordered surface. The thesis is divided into two parts. In the former we deal with the problem of combinatorial rigidity of arc complexes. In the latter we study some large-scale properties of the arc complex and the 1-skeleton of its dual, the so-called ideal triangulation graph.
