We study phase diagrams of quasi-1d systems associated with boundaries between topologically ordered states. Edges, domain walls and twist defects are all examples of such systems. In such systems, there are emergent symmetries associated with the bulk anyons which play a crucial role in the structure of their phase diagrams. Those symmetries cannot be explicitly broken by any local operator without going through a bulk phase transition. Non-local operators associated with anyons tunneling between different edges are charged under the emergent symmetries, and can serve as order parameters for the phases of the quasi-1d system. Anyon permuting symmetries (twist defects) of the bulk generate duality transformations relating different phases of the boundary and constraining the phase diagram. Many familiar phenomena can be understood in those terms. For example, edge condensation transitions in abelian states can be understood in terms of symmetry breaking or SPT transitions for the emergent symmetries. More exotic phenomena occur when the topologically ordered bulk hosts non-abelian anyons. To demonstrate these principles, we explore the boundary phase diagrams in a single and a bilayer of the toric code and the phase diagram of a domain wall in the single and the bilayer 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 duality as a symmetry of the effective edge 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.