Clusters of galaxies are the largest gravitationally bound structures in the Universe.
The hot, X-ray emitting intracluster gas that pervades clusters is the stage of complex,
nongravitational phenomena. Cool cores (CCs) are a manifestation of these complex
physical processes. High-resolution X-ray observations have established that CCs dominate the local cluster
population, with an abundance of about 70%. However, tracing this fraction to the most distant - and
thus faint and small - clusters, presents serious difficulties. In this talk I introduce the physics of cool cores and outline the original 'cooling flow' problem. I present our investigation on the fraction of CCs in the most distant cluster sample (redshift z=[0.7-1.4]) currently available in the archive of the CHANDRA observatory. Through a spatial analysis based on the surface brightness properties of these clusters, and by measuring their central cooling times, we find that the majority of the high-z clusters are in an intermediate state of cooling. I discuss the impact of this result and suggest alternative methods to confirm it.