Interesting that Steven Weinberg should cite Asymptotic Safety work by Martin Reuter and Frank Saueressig. And another UV-safety article by Roberto Percacci, that appears in Oriti's new book Approaches to Quantum Gravity: Towards a New Understanding of Space, Time, and Matter. Here's a bit from page 14 of Weinberg's recent essay http://arxiv.org/abs/0903.0568 : "...But it need not lose its predictive power at high energies, if the renormalized coupling constants gn (E) at a renormalization scale E approach a ﬁxed point gn∗ as E → ∞.  This is known as “asymptotic safety.” For this to be possible, it is necessary that βn (g∗) = 0, where βn(g (E)) ≡ E dgn(E)/dE. It is also necessary that the physical coupling constants gn(E) at any ﬁnite energy lie on a trajectory in coupling constant space that is attracted rather than repelled by this ﬁxed point. There are reasons to expect that, even with an inﬁnite number of coupling parameters, the surfaces spanned by such trajectories have ﬁnite dimensionality, so such a theory would involve just a ﬁnite number of free parameters, just as for ordinary renormalizable theories. The trouble, of course, is that there is no reason to expect the gn∗ to be small, so that ordinary perturbation theory can’t be relied on for calculations in asymptotically safe theories. Other techniques such as dimensional continuation, 1/N expansions, and lattice quantization have provided increasing evidence that gravitation may be part of an asymptotically safe theory.  So it is just possible that we may be closer to the ﬁnal underlying theory than is usually thought." Here are the references  which Weinberg cites:  M. Reuter and F. Saueressig, 0708.1317; R. Percacci, in Approaches to Quantum Gravity: Towards a New Understanding of Space, Time, and Matter, ed. D. Oriti (Cambridge Univ. Press) [0709.3851]; D. F. Litim, 0810.3675; and earlier references cited therein. Prepublication copies of Oriti's book are already in stock and on sale from several dealers http://www.amazon.co.uk/gp/offer-listing/0521860458/ref=dp_olp_new https://www.amazon.com/gp/offer-listing/0521860458/ref=dp_olp_new although the main amazon does not have it in stock yet.