Gravitational waves are predicted by the general theory of relativity. Several experiments aim at detecting them in the near future. We can think of gravitational waves as fluctuations of the curvature of the space-time manifold. They propagate at the speed of light along null hypersurfaces. In this talk, I show how by geometric-analytic methods we can lay open the structures of the relevant manifolds, and in particular of the asymptotics of the null hypersurfaces. Thereby, I explain the geometric picture of gravitational radiation and discuss also a specific nonlinear phenomenon. Finally, we relate the mathematical findings to experiments.