Ferroelectric and ferroelastic domain walls are two-dimensional (2D) topological defects with thicknesses approaching the unit cell level. When this spatial confinement is combined with observations of emergent functional properties, such as polarity in non-polar systems or electrical conductivity in otherwise insulating materials, it becomes clear that domain walls represent a new and exciting state of matter. In this review, we discuss the exotic polarisation profiles that can arise at domain walls with multiple order parameters and the different mechanisms that lead to domain wall polarity in non-polar ferroelastic materials. The emergence of energetically degenerate variants of the domain walls themselves suggests the existence of interesting quasi-1D topological defects within such walls. We also provide an overview of the general notions which have been postulated as fundamental mechanisms responsible for domain wall conduction in ferroelectrics. We then discuss the prospect of combining domain walls with transition regions observed at phase boundaries, homo- and heterointerfaces, and other quasi-2D objects, enabling emergent properties beyond those available in today’s topological systems.