I will review the known integrable structures of (super-)gravity in the presence of a sufficient number of Killing vectors and how they can be used to generate new solutions. The associated inverse scattering techniques have been widely applied to four-dimensional Einstein gravity, starting with the work of Belinski and Zakharov, but their generalisation to other gravity systems poses some problems. I will discuss progress for the case of STU supergravity by exploiting the underlying group theory.