  This article provides an introduction to the asymptotic analysis of
covariance parameter estimation for Gaussian processes. Maximum likelihood
estimation is considered. The aim of this introduction is to be accessible to a
wide audience and to present some existing results and proof techniques from
the literature. The increasing-domain and fixed-domain asymptotic settings are
considered. Under increasing-domain asymptotics, it is shown that in general
all the components of the covariance parameter can be estimated consistently by
maximum likelihood and that asymptotic normality holds. In contrast, under
fixed-domain asymptotics, only some components of the covariance parameter,
constituting the microergodic parameter, can be estimated consistently. Under
fixed-domain asymptotics, the special case of the family of isotropic Mat\'ern
covariance functions is considered. It is shown that only a combination of the
variance and spatial scale parameter is microergodic. A consistency and
asymptotic normality proof is sketched for maximum likelihood estimators.
