The identification of extended sources of X-ray emission associated with clusters of galaxies has been one of the major achievements of X-ray astronomy. In this paper the present situation regarding the identification of X-ray clusters is reviewed and the relationship between X-ray and optical cluster properties is briefly discussed. Measurements of X-ray extent have been made for a small number of clusters and the status of these observations is described. The nature of the X-ray emission process and the observations which have led to our present understanding of cluster X-ray production are examined in some detail. Observations at other wavelengths that are relevant to this problem are also discussed. The available evidence suggests that the X-ray emission is due to bremsstrahlung from high-temperature (T>2·107 K) gas that is present in the space between cluster galaxies. Current models of the origin of this gas and of its likely heating mechanism are presented.