The study of molecular Rydberg states is a rapidly growing interdisciplinary field. This valuable report, constituting the proceedings of a Royal Society Discussion reflects the state of the present theoretical knowledge, and includes an introductory chapter for the non-specialist. e material is organised to bring out the wealth of observable effects. Some states commonly lead to competitive predissociation and autoionization, with relevance to atmospheric and astrophysical processes. Others, with higher angular momenta, have much longer lifetimes and high electronic degeneracies, which on one hand pose subtle problems for experimental interpretation, and on the other act as reservoirs for a class of remarkably stable so-called ZEKE states. New experiments to classify the formation mechanism of such states, and to exploit their properties, are described. The interplay between "quantum defect" and "ab initio" aspects of the theory is also underlined.