One of the most important challenges in the modern theory of nonlinear dispersive PDEs is to understand the structure of global solutions for large times. During the last 30 years, a universal picture has emerged: asymptotically, solutions decompose in a combination of coherent structures, each with its own dynamical features, plus a dispersive remainder. Despite being a difficult and ambitious conjecture, recent progress has shown that it is not beyond reach, at least for several special important models. These advances required a combination of tools from the most diverse areas of mathematics, including nonlinear Fourier analysis, geometry, spectral theory and nonlinear elliptic PDEs. The purpose of the Conference is to gather the leading experts in the field to update on recent developments and to outline the key questions in future research.