I will report on joint work with H. Bursztyn and A. Cabrera, on which we introduce double Lie bialgebroids, which arise as second-order infinitesimal counterparts of Poisson double groupoids. I will present our results describing how these structures are related by differentiation and integration. Finally, as an application, I will discuss our approach to the categorification of Lie bialgebras by means of Poisson double structures.