The theme of the first Abel Symposium was operator algebras in a wide sense. In the last 40 years operator algebras have developed from a rather special discipline within functional analysis to become a central field in mathematics often described as non-commutative geometry (see for example the book "Non-Commutative Geometry" by the Fields medalist Alain Connes). It has branched out in several subdisciplines and made contact with other subjects like for example mathematical physics, algebraic topology, geometry, dynamical systems, knot theory, ergodic theory, wavelets, representations of groups and quantum groups. Norway has a relatively strong group of researchers in the subject, which contributed to the award of the first symposium in the series of Abel Symposia to this group. The contributions to this volume give a state-of-the-art account of some of these subdisciplines.