ar X iv : h ep - p h / 06 07 20 9 v 2 12 O ct 2 00 6 On the running coupling constant in QCD

We try to review the main current ideas and points of view on the running coupling constant in QCD. We begin by recalling briefly the classic analysis based on the Renormalization Group Equations with some emphasis on the exact solutions for a given number of loops, in comparison with the usual approximate expressions. We give particular attention to the problem of eliminating the unphysical Landau singularities, and of defining a coupling that remains significant at the infrared scales. We consider various proposal of couplings directly related to the quark-antiquark potential or to other physical quantities (effective charges) and discuss optimization in the choice of the scale parameter and of the renormalization scheme. Our main focus is, however, on dispersive methods, their application, their relation with non-perturbative effects. We try also to summarize the main results obtained by Lattice simulations in particular in various MOM schemes. We conclude briefly recalling the traditional comparison with the experimental data.