  The Jordan-Wigner transformation is known as a powerful tool in condensed
matter theory, especially in the theory of low-dimensional quantum spin
systems. The aim of this chapter is to review the application of the
Jordan-Wigner fermionization technique for calculating dynamic quantities of
low-dimensional quantum spin models. After a brief introduction of the
Jordan-Wigner transformation for one-dimensional spin one-half systems and some
of its extensions for higher dimensions and higher spin values we focus on the
dynamic properties of several low-dimensional quantum spin models. We start
from the famous s=1/2 XX chain. As a first step we recall well-known results
for dynamics of the z-spin-component fluctuation operator and then turn to the
dynamics of the dimer and trimer fluctuation operators. The dynamics of the
trimer fluctuations involves both the two-fermion (one particle and one hole)
and the four-fermion (two particles and two holes) excitations. We discuss some
properties of the two-fermion and four-fermion excitation continua. The
four-fermion dynamic quantities are of intermediate complexity between simple
two-fermion (like the zz dynamic structure factor) and enormously complex
multi-fermion (like the xx or xy dynamic structure factors) dynamic quantities.
Further we discuss the effects of dimerization, anisotropy of XY interaction,
and additional Dzyaloshinskii-Moriya interaction on various dynamic quantities.
Finally we consider the dynamic transverse spin structure factor
$S_{zz}({\bf{k}},\omega)$ for the s=1/2 XX model on a spatially anisotropic
square lattice which allows one to trace a one-to-two-dimensional crossover in
dynamic quantities.
