In a seminal paper from 2011, Wilfried Schmid and Kari Vilonen made a bold proposal that apparently intractable questions in the representation theory of real reductive groups could be resolved with the help of natural Hodge structures coming from geometry. In this talk, I will survey to what extent this program has been realised, explaining places in representation theory where Hodge structures can indeed be seen to play an important role (such as Vogan’s theory of lowest K-types, unitarity questions and the Orbit Method) and some of the techniques that go into making this a working theory. This is joint work with Kari Vilonen and with Lucas Mason-Brown.