Structured systems described by state-space models are considered. For these systems, the entries of the state-space model matrices are supposed to be either fixed zeros or free independent parameters. For such systems, one can study structural properties, i.e. properties that are valid for almost any value of the parameters. We revisit the classical geometric theory in the context of structured systems. In particular, the well-known notions of (A, B)-invariant and (C, A)-invariant subspaces are redefined and analysed in this context. We characterize fixed subspaces included or containing these invariant subspaces, and show that they play a key role in solving control problems. As an application of this geometric approach for structured systems, we propose new solutions to the disturbance decoupling problems by state feedback and by output measurement feedback. The solvability conditions are easily checked on the associated graph of the system.