Topological defects and smooth excitations determine the properties of systems showing collective order. We introduce a generic non-singular field theory that comprehensively describes defects and excitations in systems with O(n) broken rotational symmetry. Within this formalism, we explore fast events, such as defect nucleation/annihilation and dynamical phase transitions where the interplay between topological defects and non-linear excitations is particularly important. To highlight its versatility, we apply this formalism in the context of Bose-Einstein condensates, active nematics, and crystal lattices.