This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on the use of Wasserstein distances and tools from optimal transport to analyse such data. In particular, we highlight the benefits of using the notions of barycenter and geodesic PCA in the Wasserstein space for the purpose of learning the principal modes of geometric variation in a dataset. In this setting, we discuss existing works and we present some research perspectives related to the emerging field of statistical optimal transport.