Understandably, most of our intuition about nodal sets of eigenfunctions comes from landmark results on Laplace eigenfunctions. It is remarkable that these very basic intuitions can fail dramatically under relatively minor modifications of the underlying PDEs. We will see some examples of this phenomenon that arise naturally in the study of the Steklov and clamped plate problems. This talk is based on joint work with Josef Greilhuber, Angela Pistoia, and Luigi Provenzano
