This dissertation deals with some recent notions of linear dynamics of subspaces. In the first part, we provide a detailed study of n-supercyclicity and strong n-supercyclicicty in the finite dimensional setting. In particular we give a characterisation of the indices for which there exist n-supercyclic operators. We focus then on spectral properties of strongly n-supercyclic operators and on general properties as well. We also provide examples of operators whose supercyclic and strongly n-supercyclic behaviour are different. We introduce a new class of operators dealing with orbits of subspaces of finite codimension and we exhibit a \dual\ link with strong n-supercyclicity. Independently of these results, we give a characterisation of chaotic weighted shifts on a class of sequence spaces not necessarily admitting an unconditional basis. We conclude with a study of supercyclicity for unbounded operators and a sufficient condition to obtain multiple mixing operators.
