In 1993, the authors published their book “Invariant Distances and Metrics in Complex Analysis”, in which they discussed the state of affairs in the domain covered by the title of that book. In the meantime, some open questions mentioned in the book have been solved, more explicit formulas for different invariant functions on certain concrete domains were found (see also [Kob 1998]). Moreover, the classical Green function became important in studying Bergman completeness which finally led to the result that any hyperconvex bounded domain is Bergman complete. Simultaneously, a new development started, namely the study of the Green functions with multipoles. This led to the creation of a lot of new objects. Recently, there was the surprising example of the
symmetrized bidisc which initiated a lot of new activities. The symmetrized bidisc is not biholomorphically equivalent to a convex domain but, nevertheless, its Carath´eodory distance and its Lempert function coincide. Hence, it becomes again an interesting question, for which domains these two objects are equal.