  I review recent observational progress in the search for and study of distant
galaxy clusters in the X-ray band, with particular emphasis on the evolution of
the abundance of X-ray clusters out to z~1. Several on-going deep X-ray surveys
have led to the discovery of a sizeable population of clusters at z>0.5 and
have the sensitivity to detect clusters beyond redshift one. These surveys have
significantly improved our understanding of cluster evolution by showing that
the bulk of the population of galaxy clusters is not evolving significantly
since at least z~=0.8, with some evolution limited to only the most luminous,
presumably most massive systems. Thus far, a well defined sample of very high
redshift (z>~1) clusters has been difficult to assemble and represents one of
the most challenging observational tasks for the years to come.
