2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66074The connection between Riemann surfaces with boundaries and the theory of vertex operator algebras is discussed in the framework of conformal field theories defined by Kontsevich and Segal and in the framework of their generalizations in open string theory and boundary conformal field theory. We present some results, problems, conjectures, their conceptual implications and meanings in a program to construct these theories from representations of vertex operator algebras.26 pages. Final version to appear in Vertex Operator Algebras in Mathematics and Physics, Fields Institute Communications, Amer. Math. Soc., Providence. One more reference is added and some more misprints are correctedQuantum AlgebraHigh Energy Physics - TheoryMathematical PhysicsComplex Variables17B69; 81T40; 14H15, 30F10, 32G15, 81T30text