A partition of the Euclidean space into a finite number of regions is said to be "locally isoperimetric" if any variation with compact support which does not change the measure of the regions would increase the surface area of the interface. It generalizes the concept of isoperimetric sets and isoperimetric clusters to the case when some of the regions are not bounded. The problem of finding locally isoperimetric partitions was introduced by Lia Bronsard and collaborators, who also describe the first example of locally isoperimetric partition in the plane. In a joint work with Novaga and Tortorelli we prove a closure theorem which allows us to exhibit many example of locally isoperimetric partitions in the plane and in higher dimension. Moreover we develop an existence result in the planar case.