  We review the present knowledge for $\alpha_s$, the fundamental coupling
underlying the interactions of quarks and gluons in QCD. The dependence of
$\alpha_s(Q^2)$ on momentum transfer $Q$ encodes the underlying dynamics of
hadron physics -from color confinement in the infrared domain to asymptotic
freedom at short distances. We review constraints on $\alpha_s(Q^2)$ at high
$Q^2$, as predicted by perturbative QCD, and its analytic behavior at small
$Q^2$, based on models of nonperturbative dynamics. In the introductory part of
this review, we explain the phenomenological meaning of $\alpha_s$, the reason
for its running, and the challenges facing a complete understanding of its
analytic behavior in the infrared domain. In the second, more technical, part
of the review, we discuss the behavior of $\alpha_s(Q^2)$ in the high $Q^2$
domain of QCD. We review how $\alpha_s$ is defined, including its
renormalization scheme dependence, the definition of its renormalization scale,
the utility of effective charges, as well as Commensurate Scale Relations which
connect the various definitions of $\alpha_s$ without renormalization-scale
ambiguity. We also report recent measurements and theoretical analyses which
have led to precise QCD predictions at high energy. In the last part of the
review, we discuss the challenge of understanding the analytic behavior
$\alpha_s(Q^2)$ in the infrared domain. We also review important methods for
computing $\alpha_s$, including lattice QCD, the Schwinger-Dyson equations, the
Gribov-Zwanziger analysis and light-front holographic QCD. After describing
these approaches and enumerating their conflicting predictions, we discuss the
origin of these discrepancies and how to remedy them. Our aim is not only to
review the advances in this difficult area, but also to suggest what could be
an optimal definition of $\alpha_s$ in order to bring better unity to the
subject.
