High order corrections to the Renormalon
  The high order corrections to renormalon are considered. Each new type of insertions into the renormalon chain of graphs generates the correction to the asymptotics of perturbation theory of the order of $\sim 1$. However, this series of corrections to the asymptotics is not the asymptotic one (i.e. the $m$-th correction does not grow like $m!$). The summation of these corrections for UV renormalon may change the asymptotics by factor $N^\delta$. For the traditional IR renormalon the $m$-th correction diverges like $(-2)^m$. However, this divergency has no infrared origin and may be removed by proper redefinition of IR renormalon. On the other hand for IR renormalons in hadronic event shapes one should naturally expect these multi-loop contributions to decrease like $(-2)^{-m}$. Some problems expected upon reaching the best accuracy of perturbative QCD are also discussed. 
S. V. Faleev (BINP, Novosibirsk), P. G. Silvestrov (BINP, Novosibirsk   and NBI, Copenhagen)