  Considerable progress has been made over the last decade in the study of the
evolutionary trends of the population of galaxy clusters in the Universe. In
this review we focus on observations in the X-ray band. X-ray surveys with the
ROSAT satellite, supplemented by follow-up studies with ASCA and Beppo-SAX,
have allowed an assessment of the evolution of the space density of clusters
out to z~1, and the evolution of the physical properties of the intra-cluster
medium out to z~0.5. With the advent of Chandra and Newton-XMM, and their
unprecedented sensitivity and angular resolution, these studies have been
extended beyond redshift unity and have revealed the complexity of the
thermodynamical structure of clusters. The properties of the intra-cluster gas
are significantly affected by non-gravitational processes including star
formation and Active Galactic Nucleus (AGN) activity. Convincing evidence has
emerged for modest evolution of both the bulk of the X-ray cluster population
and their thermodynamical properties since redshift unity. Such an
observational scenario is consistent with hierarchical models of structure
formation in a flat low density universe with Omega_m=0.3 and sigma_8=0.7-0.8
for the normalization of the power spectrum. Basic methodologies for
construction of X-ray-selected cluster samples are reviewed and implications of
cluster evolution for cosmological models are discussed.
