In this talk I will present an approach to study nonlinear degenerate parabolic type equations and systems via their gradient flow structure in the space of Borel probability measures equipped with the so-called Wasserstein distance arising in the Monge-Kantorovich optimal transport problem. Such ideas go back to the works of Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré. In the second half of the talk --based on a joint work with I. Kim (UCLA) --, I will focus on a particular example, a so-called cross-diffusion model which involves two densities with two different drift velocities. In a particular one dimensional framework, we are able to show that the densities are guaranteed to be segregated if they were so initially, and therefore a stable interface appears between them. I will also present the incompressible limit of the system, which addresses transport under a height constraint on the total density. This leads to a two-phase Hele-Shaw/Muskat type flow.