Title: The geometry of spaces of rational curves on hypersurfaces Abstract: (From the corresponding preprint -- but may not correspond exactly to the talk) ``We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree d in Pn are not uniruled if (n + 1)/2 <= d <= n - 3. We also show that for any e >= 1, the space of smooth rational curves of degree e in a general hypersurface of degree d in P^n is not uniruled when d >= e\sqrt(n).''