  A key problem in making precise perturbative QCD (pQCD) predictions is how to
set the renormalization scale of the running coupling unambiguously at each
finite order. The elimination of the uncertainty in setting the renormalization
scale in pQCD will greatly increase the precision of collider tests of the
Standard Model and the sensitivity to new phenomena. Renormalization group
invariance requires that predictions for observables must also be independent
on the choice of the renormalization scheme. The well-known
Brodsky-Lepage-Mackenzie (BLM) approach cannot be easily extended beyond
next-to-next-to-leading order of pQCD. Several suggestions have been proposed
to extend the BLM approach to all-orders. In this paper we discuss two distinct
methods. One is based on the "Principle of Maximum Conformality" (PMC), which
provides a systematic all-orders method to eliminate the scale- and scheme-
ambiguities of pQCD. The PMC extends the BLM procedure to all orders using
renormalization group methods; as an outcome, it significantly improves the
pQCD convergence by eliminating renormalon divergences. An alternative method
is the "sequential extended BLM" (seBLM) approach, which has been primarily
designed to improve the convergence of pQCD series. The seBLM, as originally
proposed, introduces auxiliary fields and follows the pattern of the
$\beta_0$-expansion to fix the renormalization scale. However, the seBLM
requires a recomputation of pQCD amplitudes including the auxiliary fields; due
to the limited availability of calculations using these auxiliary fields, the
seBLM has only been applied to a few processes at low-orders......
