A notion of symmetry is fundamental in mathematics and physics. A new type of "quantum" symmetry has emerged in operator algebras, quantum groups, quantum invariants in low dimensional topology, integrable systems, vertex operator algebras, quantum field theory and condensed matter physics since 1980's.  I will present recent development in these areas from an operator algebraic viewpoint.  Emphasis is given to representation theoretic aspects of the Jones theory of subfactors, chiral conformal field theory and two-dimensional topological order.