The theme of the first Abel Symposium was operator algebras in a wide sense and this is the first volume in a prestigious new book series linked to the Abel prize. 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 sub-disciplines and made contact with other subjects. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topics reflect to some extent how the subject developed. 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.