  The question of boundary conditions in conformal field theories is discussed,
in the light of recent progress. Two kinds of boundary conditions are examined,
along open boundaries of the system, or along closed curves or ``seams''.
Solving consistency conditions known as Cardy equation is shown to amount to
the algebraic problem of finding integer valued representations of (one or two
copies of) the fusion algebra. Graphs encode these boundary conditions in a
natural way, but are also relevant in several aspects of physics ``in the
bulk''. Quantum algebras attached to these graphs contain information on
structure constants of the operator algebra, on the Boltzmann weights of the
corresponding integrable lattice models etc. Thus the study of boundary
conditions in Conformal Field Theory offers a new perspective on several old
physical problems and offers an explicit realisation of recent mathematical
concepts.
