This article reviews some recent progress in our understanding of the
structure of Rational Conformal Field Theories, based on ideas that originate
for a large part in the work of A. Ocneanu. The consistency conditions that
generalize modular invariance for a given RCFT in the presence of various types
of boundary conditions --open, twisted-- are encoded in a system of integer
multiplicities that form matrix representations of fusion-like algebras. These
multiplicities are also the combinatorial data that enable one to construct an
abstract ``quantum'' algebra, whose 6j- and 3j-symbols contain essential
information on the Operator Product Algebra of the RCFT and are part of a cell
system, subject to pentagonal identities. It looks quite plausible that the
classification of a wide class of RCFT amounts to a classification of ``Weak
C∗- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001
--Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in
Math., Birkhauser. References and comments adde