  YM and QCD are known to be renormalizable, but not ultraviolet finite, order
by order in perturbation theory. It is a fundamental question as to whether YM
or QCD are ultraviolet finite, or only renormalizable, order by order in the
large-N 't Hooft or Veneziano expansions. We demonstrate that Renormalization
Group and Asymptotic Freedom imply that in 't Hooft large-N expansion the
S-matrix in YM is ultraviolet finite, while in both 't Hooft and Veneziano
large-N expansions the S-matrix in confining QCD with massless quarks is
renormalizable but not ultraviolet finite. By the same argument it follows that
the large-N $\mathcal{N}=1$ SUSY YM S-matrix is ultraviolet finite as well.
Besides, we demonstrate that the correlators of local gauge-invariant
operators, as opposed to the S-matrix, are renormalizable but in general not
ultraviolet finite in the large-N 't Hooft and Veneziano expansions, neither in
pure YM and $\mathcal{N}=1$ SUSY YM nor a fortiori in massless QCD. Moreover,
we compute explicitly the counterterms that arise renormalizing the large-N 't
Hooft and Veneziano expansions, by deriving in confining massless QCD-like
theories a low-energy theorem of NSVZ type, that relates the log derivative
with respect to the gauge coupling of a $k$-point correlator, or the log
derivative with respect to the RG-invariant scale, to a $k+1$-point correlator
with the insertion of $Tr F^2$ at zero momentum. Finally, we argue that similar
results hold in the large-N limit of a vast class of confining QCD-like
theories with massive matter fields, provided a renormalization scheme exists,
as for example $\overline{MS}$, in which the beta function is independent on
the masses. In particular, in both 't Hooft and Veneziano large-N expansions
the S-matrix in confining massive QCD and massive $\mathcal{N}=1$ SUSY QCD is
renormalizable but not ultraviolet finite.
