The aim of this seminar is to present some results about the isoperimetric problem in Carnot-Carathéodory spaces connected with the Heisenberg geometry. The Heisenberg group is the framework of an open problem about the shape of isoperimetric sets, known as Pansu’s conjecture. We start by studying the isoperimetric problem in Grushin spaces and Heisenberg type groups, under a symmetry assumption that depends on the dimension. We emphasize a relation between the perimeter in these two types of structure. We conclude by presenting some recent results about constant mean curvature surfaces (hence about isoperimetric sets) in the Riemannian Heisenberg group, focusing our attention on the subriemannian limit.