Add to ORKGAbstract. We consider non-negative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space Rd, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t→ ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results. 1