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Review on the inclusive rare decays B -> X_s gamma and B -> X_d gamma in the Standard Model

We review the NLL QCD calculations for the branching ratio of B -> X_s gamma in the SM. In particular, we emphasize the problem related to the definition of the charm quark mass which leads to a rather large uncertainty of the NLL predictions. The various steps needed for a NNLL calculation, in which the m_c issue can be settled, is also sketched. We briefly summarize the results of a calculation of the O(alpha_s^2 n_f) corrections to BR(B -> X_s gamma), which was recently performed as a first step in the NNLL program. We then also briefly review the status of the photon energy spectrum and show the comparison with experimental data. Finally, we review the status of the CKM suppressed decay mode B -> X_d gamma.