  The presence or absense of renormalon singularities in the Borel plane is
shown to be determined by the analytic properties of the Gell-Mann - Low
function \beta(g) and some other functions. A constructive criterion for the
absense of singularities consists in the proper behavior of the \beta function
and its Borel image B(z) at infinity, \beta(g)\sim g^\alpha and B(z)\sim
z^\alpha with \alpha\le 1. This criterion is probably fulfilled for the \phi^4
theory, QED and QCD, but is violated in the O(n)-symmetric sigma model with
n\to\infty.
