In the mathematical theory of general relativity, null hypersurfaces in Lorentzian space-times play a crucial role. In this talk, I introduce null geodesic vector fields, which are used to construct null hypersurfaces. Then the most important geometric features of such hypersurfaces are discussed, including the definitions of shear and torsion. We give an overview of the analysis of the latter. This will be used to explain gravitational radiation in the second talk. Results on structure and asymptotic behavior of null hypersurfaces yield insight into gravitational waves.