We study the effects of radiative cooling, star formation and stellar feedback on the properties and evolution of galaxy clusters using high-resolution Adaptive Mesh Refinement N-body+gasdynamics simulations of clusters forming in the $\Lambda$CDM universe. Cooling leads to the condensation of gas in the inner regions of clusters, which in turn leads to steepening of the dark matter profile. The cooling gas is replaced by the higher-entropy gas from the outer regions, which raises the entropy and temperature of gas in the cluster core. The magnitude of these effects is likely overestimated in the current simulations because they suffer from the overcooling problem: a much larger fraction of baryons is in the form of cold gas and stars than is observed. We find that the thermal stellar feedback alone does not remedy this problem. Additional ad-hoc preheating can lower the amount of cold gas but a simple uniform preheating results in incorrect star formation history, as it delays the bulk of star formation until $z<1$. Our analysis shows that the overcooling in a cluster as a whole is really the overcooling in the central galaxy and its progenitors at high redshifts. This indicates that an additional heating mechanism that can continuously heat the gas in the cluster core is required to reproduce the observed cluster properties. Energy injection by the Active Galactic Nuclei, which may provide such heating, may thus be an important missing ingredient in the current theoretical models of cluster formation.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html