  We study effectively one-dimensional systems that emerge at the edge of a
two-dimensional topologically ordered state, or at the boundary between two
topologically ordered states. We argue that anyons of the bulk are associated
with emergent symmetries of the edge, which play a crucial role in the
structure of its phase diagram. Using this symmetry principle, transitions
between distinct gapped phases at the boundaries of Abelian states can be
understood in terms of symmetry breaking transitions or transitions between
symmetry protected topological phases. Yet more exotic phenomena occur when the
bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the
phase diagrams of the edges of a single and a double layer of the toric code,
as well as those of domain walls in a single and double-layer Kitaev spin
liquid (KSL). In the case of the KSL, we find that the presence of a
non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a
symmetry of the effective boundary theory. These examples illustrate a number
of surprising phenomena, such as spontaneous duality-breaking, two-sector phase
transitions, and unfreezing of marginal operators at a transition between
different gapless phases.
