Physicists are able to model the world with remarkable accuracy: from the largest cosmological structures to human-scale condensed-matter systems, all the way down to subatomic particles. However, in many cases the mathematical description underlying such systems is so involved that it is only possible to make predictions in an approximation where the interactions are weak. That is, one starts from a free theory and then switches interactions on, order by order in a small parameter. Not only does this strategy fail when applied to strongly-interacting systems, but it also prevents us from answering fundamental questions in theoretical physics. This Action aims at developing a comprehensive approach for studying strongly-interacting systems in classical and quantum physics by exploiting symmetries, dualities, and the internal consistency of the underlying theories. This calls for the cooperation of researchers from different fields, across Europe and beyond. The Action will bring together theoretical and mathematical physicists with expertise in quantum field theory, string theory, gravity, geometry and information theory, establishing the first network of this kind centred around Europe. Such a critical mass will also boost the visibility and impact of European research in theoretical physics. In parallel, this Action will strive to bring cutting-edge research in theoretical physics to the general public and to high-school students in particular. Working with teachers and local public bodies, it will combat long-standing prejudices on physics and research that turn away bright young students, and that in the long run may fuel indifference if not distrust towards science.