  Galaxy clusters, the largest clearly defined objects in our Universe, are
ideal laboratories to study in detail the cosmic evolution of the intergalactic
intracluster medium (ICM) and the cluster galaxy population. For the ICM, which
is heated to X-ray radiating temperatures, X-ray spectroscopy is the most
important tool to obtain insight into the structure and astrophysics of galaxy
clusters. In this review we recall the basic principles of the interpretation
of X-ray spectra from a hot, tenuous plasma and we illustrate the wide range of
scientific applications of X-ray spectroscopy. The determination of galaxy
cluster masses, important for cosmology, rest crucially on a precise
spectroscopic ICM temperature determination. The study of the thermal structure
of the ICM provides a very interesting fossil record of the energy release
during galaxy formation and evolution. The temperature and pressure
distribution of the ICM gives us important insight into the process of galaxy
cluster merging and the dissipation of the merger energy in form of turbulent
motion. Cooling cores in the centers are interesting laboratories to
investigate the interplay between gas cooling, star- and black hole formation
and energy feedback, which is diagnosed by means of X-ray spectroscopy. The
element abundances deduced from X-ray spectra of the ICM provide a cosmic
history record of the contribution of different supernovae to the
nucleosynthesis of heavy elements and their spatial distribution partly
reflects important transport processes in the ICM. Some discussion of plasma
diagnostics for conditions out of thermal equilibrium and an outlook on the
future prospects of X-ray spectroscopic cluster studies complete our review.
