abstract: The Kantorovich-Wasserstein distances between probability measures provide important examples of Optimal Transport problems: they carry remarkable geometric information of the underlying space and can be applied to many variational and PDE problems. Much less is known about (geometrically relevant) distances between positive measures with possibily different masses. We will show that effective solutions can be obtained in the new class of Optimal Entropy-Transport problems, by relaxing the marginal constraints typical of Optimal Transport and introducing a couple of convex entropies (also called Csiszar f-divergence), that penalize the deviation of the marginals of the transport plan from the assigned measures. In this way one can define a new Hellinger-Kantorovich distance between measures in met- ric spaces: it enjoys many geometric, dynamic and variational properties and lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances. (In collaboration with Alexander Mielke and Matthias Liero).