A graph is planar if it can be drawn in the plane in such a way that no two edges meet except at a vertex with which they are both incident. Any such drawing is a plane drawing of . A graph is nonplanar if no plane drawing of exists. Trees, path graphs, and graphs having less than five vertices are planar. Although since as early as 1930 a criterion for a graph to be planar was known (Kuratowski's theorem), it turned out to be difficult to use. In this Demonstration we have chosen a selection of graphs for you to test for planarity. They form a small selection of the many available in Mathematica 8. By dragging the vertices, either conjecture that it is nonplanar or try to unravel a chosen graph to get a planar drawing (if a graph is planar, this procedure will be successful, as any planar graph has a drawing in which every edge is represented by a straight line). The purpose of this Demonstration is to give you an intuitive feeling for the complexity behind a method to automatically test for planarity.