  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 with some emphasis on the exact solutions of
the RG equation 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 proposals
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 RS. 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
various MOM schemes. We conclude briefly recalling the traditional comparison
with the experimental data.
