This monograph has been given a rather broad general title, yet it is constructed around the central concepts of controllability and observability. The work does represent an investigation in the true inductive spirit from the particular to the general. The use of an interactive APL facility has allowed the work to proceed in the classical empirical manner, which demonstrates the role of a 'computer laboratory' in modern systems analysis. These lecture notes will interest those who like to learn from numerical illustrations and are interested in the scaffolding, which is so often carefully demolished in other publications.