An expansion of the real field is said to be o-minimal if every definable set has finitely many connected components. Such structures are a natural setting for studying "tame" objects of real analytic geometry such as nonoscillatory trajectories of real analytic planar vector fields. More generally, o-minimality is preserved under expanding an o-minimal structure by nonoscillatory trajectories of definable planar vector fields. But what happens when o-minimality is not preserved? In some cases, we see the best behavior that one could reasonably expect, while in others the worst possible, and we do not know at present of any other outcomes. I will make all this precise in a survey of the current state of the art.