Until very recently, topological data analysis and topological inference methods mostly relied on deterministic approaches. The major part of this habilitation thesis presents a statistical approach to such topological methods. We first develop model selection tools for selecting simplicial complexes in a given filtration. Next, we study the estimation of persistent homology on metric spaces. We also study a robust version of topological data analysis. Related to this last topic, we also investigate the problem of Wasserstein deconvolution. The second part of the habilitation thesis gathers our contributions in other fields of statistics, including a model selection method for Gaussian mixtures, an implementation of the slope heuristic for calibrating penalties, and a study of Breiman’s permutation importance measure in the context of random forests.