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We discuss the central role played by the X-ray study of hot baryons withingalaxy clusters to reconstruct the assembly of cosmic structures and to tracethe past history of star formation and accretion onto supermassive Black Holes(BHs). We shortly review the progress in this field contributed by the currentgeneration of X-ray telescopes. Then, we focus on the outstanding scientificquestions that have been opened by observations carried out in the last yearsand that represent the legacy of Chandra and XMM: (a) When and how is entropyinjected into the inter-galactic medium (IGM)? (b) What is the history of metalenrichment of the IGM? (c) What physical mechanisms determine the presence ofcool cores in galaxy clusters? (d) How is the appearance of proto-clusters atz~2 related to the peak of star formation activity and BH accretion? We showthat a highly efficient observational strategy to address these questions is tocarry out a large-area X-ray survey, reaching a sensitivity comparable to thatof deep Chandra and XMM pointings, but extending over several thousands ofsquare degrees. A similar survey can only be carried out with a Wide-FieldX-ray Telescope (WFXT), which combines a high survey speed with a sharp PSFacross the entire FoV. We emphasize the important synergies that WFXT will havewith a number of future ground-based and space telescopes, covering from theradio to the X-ray bands. Finally, we discuss the immense legacy value thatsuch a mission will have for extragalactic astronomy at large.

Galaxy clusters and the cosmic cycle of baryons across cosmic times