We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries (or "defects") are instances of a more general notion of boundaries, that give rise to quite a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role. We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. While the classification itself reproduces results obtained in a different framework by other authors before (because the underlying mathematics turns out to be the same), the physical interpretation is quite different. Dedicated to Detlev Buchholz on the occasion of his 70th birthday.