  We give the generalization of Fractional Analytic Perturbation Theory (FAPT)
for QCD observables, recently developed both for the Euclidean and Minkowski
regions of squared momentum transfer q^2, which takes into account heavy-quark
thresholds. The original analytic approach to QCD, initiated by Jones,
Solovtsov and Shirkov, is shortly summarized. We also shortly consider the
basic aspects of FAPT and then concentrate on the accounting for the
heavy-quark thresholds problem and the construction of global version of FAPT.
We discuss what one should use as an analytic coupling in the timelike region
q^2=s>0 for the e^{+}e^{-}-annihilation and the pion form factor, and consider
applications to phenomenologically relevant processes (the factorizable part of
the pion form factor and the Higgs boson decay into a b\bar{b} pair), as well
as to the summation of perturbative series.
