  We study unimodular measures on the space $\mathcal M^d$ of all pointed
Riemannian $d$-manifolds. Examples can be constructed from finite volume
manifolds, from measured foliations with Riemannian leaves, and from invariant
random subgroups of Lie groups. Unimodularity is preserved under weak* limits,
and under certain geometric constraints (e.g. bounded geometry) unimodular
measures can be used to compactify sets of finite volume manifolds. One can
then understand the geometry of manifolds $M$ with large, finite volume by
passing to unimodular limits.
  We develop a structure theory for unimodular measures on $\mathcal M^d$,
characterizing them via invariance under a certain geodesic flow, and showing
that they correspond to transverse measures on a foliated `desingularization'
of $\mathcal M^d$. We also give a geometric proof of a compactness theorem for
unimodular measures on the space of pointed manifolds with pinched negative
curvature, and characterize unimodular measures supported on hyperbolic
$3$-manifolds with finitely generated fundamental group.
