In 1967 Payne conjectured that nodal lines of a second eigenfunction of the Dirichlet Laplacian on planar domains could not be closed. In this generality, the conjecture is still open, but over the nearly four decades that have ellapsed since it has been the starting point for most of the the work that was carried out in this area.

In this talk we shall give an overview of results originating from Payne's conjecture including recent extensions to certain classes of domains, current related conjectures, and some new lines of research that have appeared within the last few years.