Motivated by important applications to geometric control theory and PDEs, recently great attention has been devoted to the study of geometric inequalities in the context of stratified Lie groups. In the first part of the talk we will discuss some classical results and open problems related to the Sobolev type inequalities in the Heisenberg group, first established by Folland and Stein in this type of generality. In the second part, we will see some applications of these inequalities to the study of a particular class of nonlinear problems. In particular, we will present some existence results for nonlinear problems involving subelliptic operators of Marcellini’s type in the Heinsenberg group.