I will talk about the scattering problem in general relativity, and present a construction of a scattering theory resolving the problem for the linearised Einstein equations in a double null gauge against a Schwarzschild background. This is done by first constructing a scattering theory for the gauge invariant components of the linearised system via the spin $\pm2$ Teukolsky equations, and this is the subject of the first part of the talk. I will then discuss how this theory can extended to the full system. Key to this step is the identification of suitable asymptotic gauge conditions on scattering data. Here, a Bondi-adapted double null gauge is shown to provide the necessary gauge rigidity, in a manner that enables the identification of Hilbert-space isomorphism between finite energy scattering data and a suitable space of finite energy Cauchy data. In particular, the gauge conditions made on scattering data will allow for a global treatment of the BMS group, and the memory effect at past and future null infinity.