Due to its nice theoretical properties and an astonishing number of applications via optimal transport problems, probably the most intensively studied metric nowadays is the p-Wasserstein metric. Given a complete and separable metric space X and a real number p belonging to [1,∞), one defines the p-Wassersteinspace W_p(X) as the collection of Borel probability measures with finite p-th moment, endowed with a distance which is calculated by means of transport plans. The main aim of our research project is to reveal the structure of the isometry group Isom(W_p(X)). Although Isom(X) embeds naturally into Isom(W_p(X)) by push-forward, and this embedding turned out to be surjective in many cases (see e.g. ), these two groups are not isomorphic in general. Kloeckner computed in  the isometry group of the quadratic Wasserstein space over the real line. It turned out that this group is extremely rich: it contains a flow of wild behaving isometries that distort the shape of measures. Following this line of investigation, we computed Isom(W_p(R)) and Isom(W_p([0,1]) for all p in [1,∞). In this talk, I will survey first some of the earlier results in the subject, and then I will present the key results of our recent manuscript . Joint work with György Pál Gehér (University of Reading) and Dániel Virosztek (IST Austria).
 J. Bertrand and B. Kloeckner, A geometric study of Wasserstein spaces: isometric rigidity in negative curvature, International Mathematics Research Notices, 2016 (5), 1368-1386.
 B. Kloeckner, A geometric study of Wasserstein spaces: Euclidean spaces, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Tome 9 (2010) no. 2, 297-323.
 Gy. P. Gehér, T. Titkos, D. Virosztek, Isometric study of Wasserstein spaces – the real line, accepted for publication in Trans. Amer. Math. Soc. Available at https://research-explorer.app.ist.ac.at/record/7389