These notes provide a student-friendly introduction to the theory of gravitational waves in full, non-linear general relativity (GR). We aim for a balance between physical intuition and mathematical rigor and cover topics such as the Newman-Penrose formalism, electromagnetic waves, asymptotically Minkowski spacetimes, the peeling theorem, the universal structure of null infinity, the Bondi-Metzner-Sachs group, and the definition of radiative modes in linear as well as in non-linear GR. Many exercises and some explicitly calculated examples complement the abstract theory and are designed to help students build up their intuition and see the mathematical machinery at work.