Over the past decade a new approach has been introduced to the theory of systems representations. Introduced by Jan C. Willems, it is called the "behavioural" approach. One of its main features is that it is well suited for modelling interconnections of systems. In this book, the author develops representation theory from a behavioural point of view and focuses on various types of ("generalized state space") first-order representations that commonly arise in the process of modelling. It covers minimality, derives transformation groups and offers realization methods that lead directly to minimal realizations. The book further presents generalized notions of controllability indices and observability indices and gives methods to calculate these indices. The book includes a procedure to derive a standard state space description from a general first-order representation. It fortifies the reader's understanding with basic examples from electrical networks and mechanics. Mathematicians and control engineers doing research on systems that are linear, time-invariant, deterministic, and finite dimensional should find this book a firm basis for understanding both the theory and applications of this behavioural approach.