Edge-flames are, roughly speaking, the two-dimensional structures in the neighborhood of a flame-sheet with an edge. They are mathematically related to 1D reaction-diffusion waves in systems with bistable equilibrium, and unbounded edge-flames can propagate with unchanging form with positive or negative propagation speeds. Frequently they are anchored. They play an important role in a wide range of combustion configurations. In the context of the workshop theme 'solid-propellant rockets' the diffusion flames supported by heterogeneous propellants have edges and there is a nonreacting gap between each edge and the propellant surface. I shall describe what we know about the structure and dynamics of edge-flames both in the non-premixed and the premixed context.