This thesis studies Quantum Field Theories (QFTs) and many-body systems that have quantum groups as global internal symmetries. We establish that quantum groups can act as genuine global symmetries, constraining the spectrum, correlation functions and dynamics. Operators that transform naturally under this symmetry are shown to be non-local, living at the ends of topological lines that carry some novel properties. These properties are illustrated by explicit examples including spin chains and their field-theoretic continuum limit. We solve the continuum theory in two ways: first, by using constraints coming from quantum group symmetry together with conformal bootstrap techniques; second, by applying the Coulomb gas formalism. We further show that this symmetry is not an artifact of integrability: it survives as a global symmetry even when integrability is broken and the system becomes chaotic. Finally, we observe that it governs late-time hydrodynamic transport, as manifest in the slow decay of the transverse spin autocorrelation function. Altogether, these results position quantum groups as a new and robust class of symmetry governing the physics of QFTs and many-body systems.