In this talk, we present a sampling of work being done at the Geometric Media Lab. The core focus of our work is to identify mathematical approaches that can be used to represent, manipulate and transform media such as images, videos and movement data, by paying careful attention to the phenomena that generate the media or are modulated by the media. In this talk, we make the case that media may have "shape," defined more precisely in the sense of high-dimensional geometric concepts. This shape, if properly understood, may hold the clue to unlocking the power of media not just as a tool for representation, but a tool for modulating rich, purposeful experience as well. We present examples of our approach in areas as diverse as movement, image and video representations.