  We consider general curvature-invariant modifications of the Einstein-Hilbert
action that become important only in regions of extremely low space-time
curvature. We investigate the far future evolution of the universe in such
models, examining the possibilities for cosmic acceleration and other ultimate
destinies. The models generically possess de Sitter space as an unstable
solution and exhibit an interesting set of attractor solutions which, in some
cases, provide alternatives to dark energy models.
