  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.
