  We study sequences of oriented Riemannian manifolds with boundary and, more
generally, integral current spaces and metric spaces with boundary.
{\color{blue}For a metric space, we define its boundary to be the completion of
the space minus the space. We impose conditions on these spaces in order to get
Gromov-Hausdorff (GH) subconvergence. By adding some other conditions} we prove
theorems demonstrating that the Gromov-Hausdorff (GH) and Sormani-Wenger
Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in
particular the limit spaces we get are countably $\mathcal{H}^n$ rectifiable
spaces. From these we derive GH compactness theorems for sequences of
Riemannian manifolds with boundary where both the GH and SWIF limits agree.
  For sequences of Riemannian manifolds with boundary we only require
nonnegative Ricci curvature, upper bounds on volume, noncollapsing conditions
on the interior of the manifold and diameter controls on the level sets near
the boundary.
