The origins of this book go back more than twenty years when, funded by small grants from the European Union, the control theory groups from the universities of Bremen and Warwick set out to develop a course in ?nite dimensional systems t- ory suitable for students with a mathematical background, who had taken courses in Analysis, Linear Algebra and Di?erential Equations. Various versions of the course were given to undergraduates at Bremen and Warwick and a set of lecture notes was produced entitled “Introduction to Mathematical Systems Theory”. As well as ourselves, the main contributors to these notes were Peter Crouch and Dietmar Salamon. Some years later we decided to expand the lecture notes into a textbook on mathematical systems theory. When we made this decision we were not very realistic about how long it would take us to complete the project. Mathematical control theory is a rather young discipline and its foundations are not as settled as those of more mature mathematical ?elds. Its basic principles and what is c- sidered to be its core are still changing under the in?uence of new problems, new approaches and new currents of research. This complicated our decisions about the basic outline and the orientation of the book. During the period of our writing, problems of uncertainty and robustness, which had been forgotten for some time in ’modern control’, gradually re-emerged and came to the foreground of control theory.