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Conformal blocks form a system of vector bundles over the moduli space ofcomplex curves with marked points. We discuss various aspects of these bundles.In particular, we present conjectures about the dimensions of sub-bundles. Theyimply a Verlinde formula for non-simply connected groups like PGL(n,C).  We then explain how conformal blocks enter in the construction of conformalfield theories on surfaces with boundaries. Such surfaces naturally appear inthe conformal field theory description of string propagation in the backgroundof a D-brane. In this context, the sub-bundle structure of the conformal blockscontrols the structure of symmetry breaking boundary conditions.

D-Brane Conformal Field Theory and Bundles of Conformal Blocks