The Periodic Table effectively embraces the whole realm of chemistry within the confines of one comparatively simple and easily understood chart of the chemical elements. Over many years the Periodic Table has proven to be indispensable not only to chemists of all kinds but also to a host of other scientists, including biologists, geologists and physicists. It is thus hardly surprising that the Periodic Table has become one of our most celebrated contemporary scientific icons. In the present work various aspects of the Periodic Table that are seldom if ever featured elsewhere are given prominence. The twelve presentations contained herein all have a mathematical flavour because it is the intention to highlight the often-neglected mathematical features of the Periodic Table and several closely related topics. The book starts out by considering predictions of what the ultimate size of the Periodic Table will be when all of the possible artificial chemical elements have been synthesised. It then moves on to an examination of the nature of the periodicity extant in the Periodic Table and some methods for the prediction of the properties of the super-heavy elements.The Periodic Table is next explored in various dimensions other than two. The natural clustering of the elements into groups is studied by three different but complementary routes, namely via the topological structures of the groups, the self-association of the elements as evidenced by neural network studies, and information theoretical analysis of the behaviour of atoms. Following a detailed investigation of the mathematical basis for the periodicity seen in atomic and molecular spectroscopy, three separate presentations delve into many different aspects of the group-theoretical structure of the Periodic Table. The unusual combination of themes offered here will appeal to all who seek a more detailed and intimate knowledge of the Periodic Table than that available in standard texts on the subject.