Abstract: Signals are used in our everyday life to send and receive information or to extract information from an unknown environment. Typically, signals are defined over a metric space, i.e. time and space. The goal of this talk is to present a set of tools to analyze signals that are defined over a topological (e.g., not necessarily metric) space, i.e. a set of points along with a set of neighbourhood relations for each point. Motivating applications span from gene regulatory networks to social networks, etc. We introduce the Graph Fourier Transform (GFT), derive an uncertainty principle for signals defined over graphs and set the basis for a sampling theory over graphs. We start from signals defined over graphs and then we move to the most general case of signals defined over simplicial complexes. Finally, we illustrate some applications to the recovery of the electromagnetic field from a subset of observations, the inference of the brain functional activity network from electrocorticography (ECoG) signals collected in an epilepsy study and theprediction of data traffic over telecommunication networks.