The development, by Wonham and co-workers, of the concept of a controllability subspace and elucidation of its relationship with pole assignment have profoundly influenced the geometric control theory of state-space systems. This paper develops the corresponding theory for linear systems described by a mixture of algebraic and differential equations. The proofs given use eigenstructure assignment and include necessary and sufficient conditions for the assignability of a given eigenstructure, including the so-called infinite eigenstructure. This approach provides some simplifications of the state-space theory, such as an intrinsic characterization of controllability subspaces, and a very general result concerning finite pole assignment in descriptor systems. Throughout, care is taken to ensure the existence and uniqueness of classical solutions of the underlying differential equations, which cannot be taken for granted.