  The regularity of limit spaces of Riemannian manifolds with L^p curvature
bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption.
A regular subset, defined by a local volume growth condition for a limit
measure, is shown to carry the structure of a Riemannian manifold. One
consequence of this is a compactness theorem for Riemannian manifolds with
$L^p$ curvature bounds and an a priori volume growth assumption in the pointed
Cheeger-Gromov topology. A different notion of convergence is also studied,
which replaces the exhaustion by balls in the pointed Cheeger-Gromov topology
with an exhaustion by volume non-collapsed regions. Assuming in addition a
lower bound on the Ricci curvature, the compactness theorem is extended to this
topology. Moreover, we study how a convergent sequence of manifolds disconnects
topologically in the limit. In two dimensions, building on results of Shioya,
the structure of limit spaces is described in detail: it is seen to be a union
of an incomplete Riemannian surface and 1-dimensional length spaces.
