Simple groups are notoriously complicated objects to study. The key is generally to find the right geometrical or combinatorial structure for the group to act on. I will give an overview of the types of structure that are often used, and introduce some new structures that make certain simple groups more accessible. While the emphasis is on finite simple groups, there are also applications to simple Lie groups, especially E7 and E8, and from there into physics. (Professor Wilson is a collaborator of Corinne Manogue and Tevian Dray.)