A geometric representation for multivariate extremes, based on the shapes of scaled sample clouds in light-tailed margins and their so-called limit sets, has recently been shown to connect several existing extremal dependence concepts. Furthermore, this geometric representation provides a natural way to describe complex extremal dependence structures, which more established approaches to multivariate extremes do not represent well. These attractive properties have led to recent work that exploits the geometric approach as a foundation for statistical modelling, which has been demonstrated in relatively low dimensions thus far. For higher dimensional modelling, we require principled simplifications of the model structure. We will introduce the concept of geometric extremal graphical models, and outline some theoretical results based on block graphs. On the practical side, we will demonstrate some initial results employing these ideas to model joint river flows in the northwest of England. Based on joint works with Ioannis Papastathopoulos, and Kristina Grolmusova and Thordis Thorarinsdottir.