Combinatorial curvatures group actions and colourings: aspects of topological combinatorics The first part asks whether classical notions of curvature known from differential geometry can be carried over to discrete structures. Of particular interest are analogies on a structural level; questions of convergence are not considered. Chapter 1 is motivated by constructions well known in differential geometry. The basic object a difference operator makes the computation of a discrete analogue of the rough Laplacian and of the Riemannian curvature tensor possible. A trace of the latter yields a discrete notion of Ricci curvature. Applications to the objects introduced in chapter 1 are discussed in chapter 2. The first application is the proof of a combinatorial formula of Weitzenböck type that coincides with a decomposition of Weitzenböck type postulated by Forman in many cases. Second a combinatorial version of the theorem due to Gauß and Bonnet is proved for combinatorial 2 manifolds. For this theorem a special choice of weights has to be assigned to the cells that yield a Weitzenböck formula different from the one postulated by Forman. The following sections discuss whether Formans combinatorial version of a theorem of Bochner for 1 chains can be generalised to the weighted notion of Ricci curvature introduced in chapter 1 ar to p chains with p>1. The chapter ends with a generalisation of Formans diameter estimate to the weighted notion of Ricci curvature and simple examples show that his approach is indeed generalised. Part 2 asks for lower bounds for the chromatic number of graphs and hypergraphs obtained using topological methods ...