  The subject of the first section-lecture is concerned with the strength and
the weakness of the perturbation theory (PT) approach, that is expansion in
powers of a small parameter $\alpha$, in Quantum Theory. We start with
outlining a general troublesome feature of the main quantum theory instrument,
the perturbation expansion method. The striking issue is that perturbation
series in powers of $\alpha \ll 1$ is not a convergent series. The formal
reason is an essential singularity of quantum amplitude (matrix element)
$C(\alpha)$ at the origin $\alpha=0$. In many physically important cases one
needs some alternative means of theoretical analysis. In particular, this
refers to perturbative QCD (pQCD) in the low-energy domain. In the second
section-lecture, we discuss the approach of Analytic Perturbation Theory (APT).
We start with a short historic preamble and then discuss how combining the
Dispersion Relation with the Renormalization Group (RG) techniques yields the
APT with \myMath{\displaystyle e^{-1/\alpha}} nonanalyticity. Next we consider
the results of APT applications to low-energy QCD processes and show that in
this approach the fourth-loop contributions, which appear to be on the
asymptotic border in the pQCD approach, are of the order of a few per mil. Then
we note that using the RG in QCD dictates the need to use the Fractional APT
(FAPT) and describe its basic ingredients. As an example of the FAPT
application in QCD we consider the pion form factor $F_\pi(Q^2)$ calculation.
At the end, we discuss the resummation of nonpower series in {(F)APT} with
application to the estimation of the Higgs-boson-decay width
$\Gamma_{H\to\bar{b}b}(m_H^2)$.
