Topological defects occur in a wide variety of mathematical and physical settings. I will review the origin, structure and dynamics of such defects including a recent realization of nonabelian line defects in a three-dimensional chiral liquid crystal system with the associated entanglement, trivalent junctions and networks, and stable bound states of pairs of defects, thus realizing the notion of topological rigidity envisaged 50 years ago by Poenaru and Toulouse.