  Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat
distance between oriented $m$ dimensional Riemannian manifolds with boundary by
isometrically embedding the manifolds into a common metric space, measuring the
flat distance between them and taking an infimum over all isometric embeddings
and all common metric spaces. This is made rigorous by applying
Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents
to arbitrary metric spaces. We prove the intrinsic flat distance between two
compact oriented Riemannian manifolds is zero iff they have an orientation
preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we
study converging sequences of manifolds and their limits, which are in a class
of metric spaces that we call integral current spaces. We describe the
properties of such spaces including the fact that they are countably
$\mathcal{H}^m$ rectifiable spaces and present numerous examples.
