  To each finitely generated group $G$, we associate a quasi-isometric
invariant called the Dehn spectrum of $G$. If $G$ is finitely presented, our
invariant is closely related to the Dehn function of $G$. The main goal of this
paper is to initiate the study of Dehn spectra of finitely generated (but not
necessarily finitely presented) groups. In particular, we compute the Dehn
spectrum of small cancellation groups, certain wreath products, and free
Burnside groups of sufficiently large odd exponent. We also address some
natural questions on the structure of the poset of Dehn spectra. As an
application, we show that there exist $2^{\aleph_0}$ pairwise
non-quasi-isometric finitely generated groups of finite exponent.
