In TDA, persistent homology appears as a fundamental tool to infer relevant topological information from data. Persistence diagrams are usually computed from filtrations built on top of data sets sampled from some unknown (metric) space. They provide "topological signatures" revealing the structure of the underlying space. To ensure the relevance of such signatures, it is necessary to prove that they come with stability properties with respect to the way data are sampled. In this talk, we will present a few results on the stability of persistence diagrams built on top of general metric spaces. We will show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties.