Persistent Homology is one of the main tools of Topological Data Analysis. It consists in comparing, by homology, all pairs of sublevel sets of a pair (X, f) where X is a topological space (or a simplicial complex) and f is a real valued function on it. This produces a Persistence Diagram, a standard object which turns very useful in shape analysis and classification. But is the topological setup necessary for getting persistence diagrams? Massimo Ferri will introduce (classical) persistent homology in the first part. Alessandro Mella will then show a wide generalization of it and some initial applications.