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 \approx 1 , and the evolution of the physical properties of the intra-cluster medium out to z \approx 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 } \simeq 0.3 and \sigma _ { 8 } \simeq 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 .