  The main purpose of this chapter is to present some theoretical aspects of
parametric estimation of L\'evy processes based on high-frequency sampling,
with a focus on infinite activity pure-jump models. Asymptotics for several
classes of explicit estimating functions are discussed. In addition to the
asymptotic normality at several rates of convergence, a uniform
tail-probability estimate for statistical random fields is given. As specific
cases, we discuss method of moments for the stable L\'evy processes in much
greater detail, with briefly mentioning locally stable L\'evy processes too.
Also discussed is, due to its theoretical importance, a brief review of how the
classical likelihood approach works or does not, beyond the fact that the
likelihood function is not explicit.
