This paper considers structural properties of non-proper linear multivariable systems in Rosenbrock's generalized state-space form. A particular form of the system matrix is shown to yield various relevant structural properties. The relationship with the theory of singular pencils of matrices and with geometric properties is investigated and provides a full explanation of Rosenbrock's definitions of the zeros of a system. New definitions are proposed for the poles, zeros and decoupling zeros at infinity of a non-proper system. The relevance and the properties of the concepts proposed here are investigated in detail and lead to generalizations of the work of Rosenbrock and Vergliese. The redundancy represented in the polynomial part of a system is established and this yields in two different ways a generalization of Kalman's canonical decomposition of the state-space. Finally, the relationship with invertibility properties is investigated.