This volume contains a collection of refereed papers by eminent experts originating from the "International Conference on Semigroups of Operators: Theory and Control", held in December 1998 in Newport Beach, California. They highlight recent advances in the theory of semigroups of operators which provide the framework for the time-domain solutions of time-invariant boundary value and initial value problems of partial differential equations. There is a firewall between the abstract theory and the applications, and one of the conference aims, which is reflected in this collection, was to bring them together for the benefit of both communities.