In this talk we will consider Dehn functions of groups. After a quick review of their interest and main properties, we'll introduce the much more modern notion of "mean Dehn function". In the second part of the talk, we'll scketch the proof of the following theorem: "the mean Dehn function of a finitely generated abelian group is )". This result makes a big contrast with the well known facts that the Dehn function of abelian groups is quadratic, and that there is no group with Dehn function between quadratic and linear. The proof consists on a detailed combinatorial analysis involving several countings of paths of given length in the integral lattice .