NOTE: This lecture is part of a course on galaxy clusters for Insubria University graduate students. Participation is open to all interested. ABSTRACT: I first review the early X-ray observations of galaxy clusters, which led to the cooling flow theory (CFT). I explore the framework of this theory, based on the observation that the intra-cluster gas often has a radiative cooling time much shorter than the cluster age, which implies a slight unbalance of the hydrostatic equilibrium, hence an inflow of mass towards the cluster core.The modern observations with the X-ray facilities Chandra and XMM-Newton did not confirm the CFT key predictions. The absence of strong cooling flows implied by the new observations opened new problems, chiefly related to the need of an efficient heating mechanism able to stifle the radiative losses. I end with a review of the recent progress in this direction.