The perennial appeal of condensed matter science lies in its apparently limitless set of possibilities. Combining different sorts of constituent particles in different ways leads to myriads of remarkable phenomena; one cannot help but wonder that of these countless possibilities, only a few are known, and the simplest understood. Our problem is that the predictive power of our standard approaches is quickly rendered inapplicable by the potent mixture of quantum mechanics, interparticle interactions and reduced dimensionality. The ubiquitous strategy of focusing on low-energy, universal features only often lacks the depth to explain or predict important, observable and potentially useful features of key strongly-correlated systems. This proposal aims to establish a new framework for tackling strong correlations, in which universal scaling results at low energies are superseded by detailed, quantitative results valid at arbitrary energies, but in more specific circumstances. This will be achieved by capitalizing on recent breakthroughs on `pioneer systems: these `Bethe liquids find experimental realizations in forms ranging from magnetic materials and cold atomic gases to quantum dots and nanostructures. Their theoretical description will be unified into a consistent framework, bridging diverse fields from the mathematics of representation theory all the way to concrete physical applications, and offering urgently-needed, detailed quantitative calculations of many experimental observables beyond the reach of traditional methods. The theoretical foundations of Bethe liquids will in fact be robust enough to support the treatment not only of equilibrium but also of out-of-equilibrium situations, allowing detailed studies of fundamental questions relating to dynamics, decoherence, relaxation and thermalization in many-body interacting quantum systems, providing stringent new benchmarks for alternative methods, setting new challenges for experiments, and providing an altogether new way of thinking about strong correlations.