The Dehn function of a finite presentation of a group $G = \langle A \mid R \rangle$ gives the least upper bound for the number of relators that must be applied to a word $w \in G$ that is trivial i.e., $w =_G 1$, in order to reduce $w$ to the trivial word. Up to a natural equivalence on functions, the Dehn function is a quasi-isometry invariant of the group $G$. The study of Dehn functions gained importance after Gromov's seminal theorem: a finitely presented group $G$ has sub-quadratic Dehn function if and only if $G$ has linear Dehn function if and only if the Cayley graph of $G$ is $\delta$-hyperbolic.

In this talk we will outline the various definitions and ideas in the subject. Then we will address the basic question: what possible functions can arise as Dehn functions of finitely presented groups? We will outline the construction of the so-called {\it snowflake groups} which give many new examples of Dehn functions. The results presented are joint work with Noel Brady, Martin Bridson and Max Forester.