The aim of this course is to present some recent advances in the theory of stable sheaves on higher dimensional varieties, in particular Fano and hyper-Kähler manifolds. We will start by reviewing the case of surfaces. We will then turn to Fano threefolds, where the theory is in a certain sense a generalization of the case of curves, by studying in detail the case of cubic threefolds. Finally, we will consider hyper-Kähler fourfolds, where the theory gives a generalization of Mukai's theory on K3 surfaces. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.