This volume presents articles from several lectures presented at the school on 'Quantum Symmetries in Theoretical Physics and Mathematics' held in Bariloche, Argentina. The various lecturers provided significantly different points of view on several aspects of Hopf algebras, quantum group theory, and noncommutative differential geometry, ranging from analysis, geometry, and algebra to physical models, especially in connection with integrable systems and conformal field theories. Primary topics discussed in the text include subgroups of quantum $SU(N)$, quantum ADE classifications and generalized Coxeter systems, modular invariance, defects and boundaries in conformal field theory, finite dimensional Hopf algebras, Lie bialgebras and Belavin-Drinfeld triples, real forms of quantum spaces, perturbative and non-perturbative Yang-Baxter operators, braided subfactors in operator algebras and conformal field theory, and generalized ($d^N$) cohomologies.
About finite dimensional Hopf algebras by N. Andruskiewitsch Lectures on differentials, generalized differentials and on some examples related to theoretical physics by M. Dubois-Violette Modular invariants from subfactors by J. Bockenhauer and D. E. Evans The classification of subgroups of quantum $SU(N)$ by A. Ocneanu Uses of quantum spaces by O. Ogievetsky CFT, BCFT, ADE and all that by J.-B. Zuber.