Topological defects are hallmarks of systems exhibiting collective order. They are widely encountered from condensed matter, including biological systems, to elementary particles, and the very early Universe. The small-scale dynamics of interacting topological defects are crucial for the emergence of large-scale non-equilibrium phenomena, such as quantum turbulence in superfluids, spontaneous flows in active matter, or dislocation plasticity in crystals. In-depth understanding and tailoring of collective behaviors require a unified description of defects associated with symmetry breaking and the non-topological excitations of ground states. Due to the highly non-linear nature, it still remains poorly understood. In this work, Vidar Skogvoll et al. from the Department of Physics, University of Oslo, presented a formalism to describe the evolution of ordered systems from the dynamics of their topological defects and their interactions with smooth but localized excitations. The versatility of the approach allows them to gain insight into defect annihilation, the onset of collective behavior, and perspectives on defect structures. In particular, the authors applied the method to systems of increasing topological and dynamical complexity. They studied the motion of isolated vortices in Bose-Einstein condensates, which, in addition to confirming that the method correctly identifies topological defects and their velocities, sheds light on changes in quantum pressure arising from the interplay between phase slips and shock waves. For active nematics, they observed that the onset of active turbulence as a melting of periodic arches is signaled by the formation of bound dipoles of nematic defects at the core of dislocations in the nematic arches. Although the whole framework is presented for systems with one broken rotational symmetry, it is a powerful tool that can be generalized to systems with multiple broken symmetries and reveal hidden hierarchies of topological defects associated with each symmetry, laying the foundation for unified theories in systems characterized by collective behaviors.