We define a new distance between oriented Riemannian manifolds that we call the "intrinsic flat distance" based upon Ambrosio-Kirchheim's theory of integral currents on metric spaces. Limits of sequence of manifolds with a uniform upper bound on their volume and diameter are countably H^m rectifiable metric spaces with an orientation and multiplicity that we call "integral current spaces". In general the Gromov-Hausdorff and intrinsic flat limits do not agree. Intrinsic flat convergence is a weaker notion. We show that they do agree when the sequence of manifolds has nonnegative Ricci curvature and a uniform lower bound on volume and also when the sequence of manifolds has a uniform linear local geometric contractibility function. These results are proven using work of Greene-Petersen, Gromov, Cheeger-Colding and Perelman. This is joint work with S. Wenger.