Given a simple graph G, is it possible to represent its vertices by simply connected regions in the plane so that two regions overlap if and only if the corresponding two vertices are adjacent? In other words, is G isomorphic to the intersection graph of a set of simply connected regions in the plane? This deceptively simple extension of propositional logic and its generalizations are often referred to in the literature as "topological inference problems." They have proved to be relevant in the area of geographic information systems, in neurology, VLSI design, and in graph drawing. We outline some recent results related to these questions.