The conformal bootstrap methods are an exciting and fast-growing area of research, constituting a definite break-away from the more traditional (and often less rigorous) ways of studying systems at criticality. They provide us with a unique tool for exploring strongly coupled theories which are otherwise inaccessible, by using the powerful constraints of conformal symmetry. The European Union has an important opportunity in establishing itself now as a world leader in this promising, burgeoning field of research, by supporting projects such as the one we propose.We will study the O(N) vector models and their defects, examining their properties at criticality from the point of view of conformal symmetry. Our goal is to provide experimentalists and computational physicists with concrete testable predictions. We also hope to clarify any special analytic properties which may underlie these models. At the same time, we will develop the conformal bootstrap methodology, both theoretically and computationally. Finally, we will develop and publish a computer package for basic bootstrap calculations. This will allow for a greater number of researchers to use and adapt our techniques for their own purposes.The research project will take place at CERN, one of the world’s most active and perhaps largest research centers. The atmosphere is a vibrant one, with a department containing dozens of permanent researchers, fellows and a large number of temporary and long-term visitors. CERN is constituted by 20 member states, of which 18 belong to the EU, together with other four candidate or associate members. Such an international constitution guarantees a steady flow of ideas and many opportunities for cross-fertilization between fields, and international collaborations.