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.