The concept of topological persistence introduced independently by several groups \cite{elz-tps-02,r-tchfa-99,f-dcsf-92} is a rather general tool providing an efficient way to encode the qualitative and quantitative behavior of real-valued functions defined over topological spaces. Since its introduction this encoding, known as the persistence diagram or barcode, has been extensively studied, specifically in topological data analysis where its stability properties allow to infer robust topological information on the studied data sets. Motivated by problems coming from topological data analysis, we extend the notions of persistence and persistence diagrams to a larger setting than the one classically considered. In this paper we prove new stability results for the persistence diagrams that lead to new applications in topological and geometric data analysis.