The m-th isoperimetric or filling volume function of a Riemannian
manifold or a more general metric space X measures how difficult it is
to fill m-dimensional boundaries in X of a given volume with an
(m+1)-dimensional surface in X. The asymptotic growth of the
isoperimetric functions provides large scale invariants of the
underlying space. They have been the subject of intense research in past
years in large scale geometry and especially geometric group theory,
where the isoperimetric functions appear as Dehn functions of a group.
In this talk, I survey relationships between the asymptotic growth of
isoperimetric functions and the large scale geometry of the underlying
space and, in particular, fine properties of its asymptotic cones. I
will furthermore describe recently developed tools from geometric
measure theory in metric spaces and explain how these can be used to
study the asymptotic growth of the isoperimetric functions.