This work summarises the salient features of current and planned experiments into multiquark hadrons, describing various inroads to accommodate them within a theoretical framework. At a pedagogical level, authors review the salient aspects of quantum chromodynamics (QCD), the theory of strong interactions, which has been brought to the fore by high-energy physics experiments over recent decades. Compact diquarks as building blocks of a new spectroscopy are presented and confronted with alternative explanations of the XYZ resonances. Ways to distinguish among theoretical alternatives are illustrated, to be tested with the help of high luminosity LHC, electron-positron colliders, and the proposed Tera-Z colliders. Non-perturbative treatments of multiquark hadrons, such as large N expansion, lattice QCD simulations, and predictions about doubly heavy multiquarks are reviewed in considerable detail. With a broad appeal across high-energy physics, this work is pertinent to researchers focused on experiments, phenomenology or lattice QCD.