Dougal Davis, of the University of Melbourne, surveys a research program begun by Wilfried Schmid and Kari Vilonen in a 2011 paper proposing that Hodge structures arising from geometry can resolve long-standing problems in the representation theory of real reductive groups. Delivered as part of the Joint IAS/PU Arithmetic Geometry seminar, the talk traces how far this program has advanced, pointing to specific areas where Hodge structures now play a visible role: Vogan's theory of lowest K-types, questions of unitarity, and the orbit method. Davis outlines some of the technical machinery needed to make the theory work, drawing on joint work with Vilonen and Lucas Mason-Brown. The talk is aimed at a research audience already acquainted with representation theory and Hodge theory, moving through the material at seminar pace rather than pausing for introductory exposition.