Abstract: Discrete homotopy theory is a way of analyzing graphs with topological ideas. Among other discrete analogs of familiar concepts, it is possible to define discrete homotopy groups and discrete homology groups of a graph which give insight into the connectivity of the graph. In the talk, an introduction to the emerging field will be given. Additionally, a notion of Cat(0) graphs will be introduced, whose theory resembles the theory of Cat(0) cubical complexes. The Cartan-Hadamard theorem for cubical complexes implies that Cat(0) cubical complexes are contractible; a combinatorial analog of this theorem for graphs will be presented. As an application, the discrete homotopy types of Cayley graphs of Coxeter groups will be determined.