After a review of various types of tilings and aperiodic materials, the notion of tiling space (or Hull) will be defined. The action of the translation group makes it a dynamical system. Various local properties, such as the notion of "Finite Local Complexity" or of "Repetitivity" will be interpreted in terms of the tiling space. The special case of quasicrystal will be described. In the second part of the talk, various tools will be introduced to compute the topological invariants of the tiling space. First the tiling space will be seen as an inverse limit of compact branched oriented manifolds (the Anderson-Putnam complex). Then various cohomologies will be defined on the tiling space giving rise to isomorphic cohomology groups. As applications, the "Gap Labeling Theorem" will be described, and some results for the main quasicrystal and simple tilings will be given.
