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. The 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...