Interpreting and comparing shapes are challenging issues in computer vision, computer graphics and pattern recognition. Persistent homology allows us to describe shapes by means of suitable shape descriptors, the persistence diagrams. They capture both geometrical and topological features of an object by recording the homological changes and their longevity in the lower level sets of a real valued function (called a filtering function) defined on a topological space associated with the object. This seminar provides an overview of some of the most relevant results in persistent homology, the main properties of persistence diagrams and the principal aspects in the generalization to multivalued filtering functions that allow the description of shape properties with an intrinsically multidimensional nature.