Several combinatorial objects (including several types of random walks) have a recursive combinatorial description that leads to a (system of) functional equation(s) for the corresponding counting generating function, where the right hand side of the equation has non-negative coefficients; sometimes there also appears a catalytic variable, for example for random walks restricted to some region or for the enumeration of planar maps. The purpose of this talk to show that the positivity condition leads to universal asymptotic properties of the underlying counting problem.