Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.