This chapter derives practical conditions for the observability of the initial state and introduces methods for determining it. It discusses continuous linear systems and examines discrete linear systems. The chapter introduces and uses an analogy between the derived observability criteria and the conditions for the complete controllability of linear systems. It examines the observability of the initial state of the continuous linear system. The chapter discusses the special case of time-invariant systems. It also examines the observability of the discrete linear system. The chapter analyzes the relation between the observability of a linear system and the controllability of its adjoint system. It presents some applications of the observability theory and duality of linear systems in engineering and in the social sciences. The chapter suggests that the construction of a dual system is very similar to that of a linear programming problem.