The Teichmüller space of a surface is the space of complex structures on this surface, and the quantum Teichmüller space is a certain deformation of the algebra of rational functions on the Teichmüller space. In spite of its geometric origins, the quantum Teichmüller space is a purely algebraic object, depending of the combinatorics of of the complex of all ideal triangulations of the surface. I will discuss some applications of its representation theory, and its conjectural connections with other combinatorial objects, such as the skein module of the surface.