  In this paper we study the evolution of almost non-negatively curved
(possibly singular) three dimensional metric spaces by Ricci flow. The
non-negatively curved metric spaces which we consider arise as limits of smooth
Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than
-c^2(i), where c^2(i) goes to zero as i goes to infinity, and whose diameter is
bounded by a constant independent of i, and whose volume is bounded from below
by a positive constant independent of i. We show for such spaces, that a
solution to Ricci flow exists for a short time, and that the solution is smooth
for all positive times and that it has non-negative Ricci curvature. This
allows us to classify the topological type and the differential structure of
the limit manifold (in view of Hamilton's Theorem on closed three manifolds
with non-negative Ricci curvature).
